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I can't speak to your ancestor's individual motivations, of course, but immediate expectation of conscription wouldn't be one of them in 1860. The United States had never drafted anybody, at least on a national scale, at that time. Doing so was a controversial step taken only after it became clear that the short, swift war with a decisive battle or two was not going to materialize. The boys, as it were, well and truly would not be back for Christmas. Nor were all the volunteers who signed on for ninety days or a year or so going to stick around and re-up. If your ancestor was lying about his age in 1864, different story.

But that leaves us the broader question: Did anybody see all this coming? The answer there that it depends what one means by "see the Civil War coming" and also what "the Civil War entails" in their minds when they do or don't see it. So let's open the box on Schroedinger's Antebellum American for the firm yes and firm no that are both correct.

Predictions of a civil war are almost as old as the Republic, regularly raised as the obvious result of an attempt at breaking the Union. These largely take the role of a general fear and warning, rather than a specific thing people expect to happen in the immediate future. So in the most permissive sense we can say that many people saw it coming, but that papers over a lot of important nuance. A theoretical civil war is, most of the time for most Americans, not a major concern. When tensions over slavery rise -and it's always slavery that brings things to this point- there's a sense that Something Must Be Done or the whole nation may explode. That Something is basically to give slavery extra protections and advantages beyond its already extraordinary protection granted by the Constitution, guarantees which considerably exceed any later provided for freedom or equality. There's usually also a fig leaf for free state pols so they can say they didn't totally betray their constituents (at least the ones who dislike slavery) but those concessions are insbustantial and never remotely commensurate with what the enslavers got.

Until the late 1840s, the steady give (to enslavers) and take (from the enslaved) is the general outline of American politics without major exception. The conflict over the spoils of the Mexican War marked an important departure from that norm and things never quite get back to the status quo ante. There's always been conflict over slavery's future in the nation, but coming into the 1850s the usual rounds of heavy capitulation to the enslavers don't do the trick anymore. Partly, it's that they've gotten so extreme that ordinary whites across the North now feel personally implicated in slavery in ways they were not before courtesy of the Fugitive Slave Act but there's also a general trend of lesser known proslavery provocations that slowly added weight to the antislavery case in the decades prior. The two trends feed on each other in important ways, but most white Americans outside the general evangelical reformist movement of the era don't consider themselves primarily antislavery in their politics until the last antebellum decade even if they do oppose the institution. The big reactions to the Compromise of 1850, followed up by the realization that it really settled nothing, a series of daring rescues of fugitive slaves from their would-be captors, and similar genuinely popular resistance to slavery kick off the decade.

The tension only ratchets up from there, with a new provocation in the form of the Kansas-Nebraska Act. The law opens vast new territories to slavery, where American law had previously forbade its entry for thirty years. That whether or not slavery will go into Kansas will be settled on the ground leads to a fair bit of actually fighting it out during small-scale violence that erupts for a while into a little war within the territory. Seeing this as their chance to put right what went wrong with the Kansas-Nebraska Act, antislavery Northerners are eager to fund settlement by their own people in the territory. Proslavery Missourians view this as stealing "their" turf and respond with semisecret paramilitaries, massive vote fraud, and a fair bit of violence around polling places. All of this keeps the issue in the news and at the center of American politics past the middle of the decade.

This is all way too much for the systems built up in the 1820s and 1830s to muzzle antislavery whites and defend slavery to handle. The biggest of those are two national parties, with enough constituency and power in each section to act as effective pressure on antislavery members in their own ranks to keep in line and to a lesser degree to force other issues which divide antislavery men between the two parties. As long as antislavery whites are divided between Whigs (mostly) and Democrats (an important minority) and other issues on which they disagree (the tariff, internal improvements, the national bank) remain the focus of politics, all this can keep puttering along. When slavery takes center stage and becomes the issue, then who cares about the bank, some roads, or a duty on whatever? In that environment, being antislavery counts more and the party coalitions fracture.

That happens in every sectional clash, which is why more conservative types understand them as so threatening. What happens in 1852-5 is a lot worse than usual because the dispute over the Mexican Cession went on for literal years and the solution kept the issue alive. That solution was a Whig program, roughly, originally managed by the party's chief founder Henry Clay. Clay's solution did not get through Congress with mainly Whig votes because the party's antislavery wing would not accept it and their proslavery wing was lukewarm at best. Instead it's managed by Stephen Douglas, the Democracy's rising star. In the South, that means that the Democrats can deliver for slavery and the Whigs cannot. Until this point, Whiggery has been the South's second party and genuinely competitive in some states. The fallout from 1850 is a lot of Whigs staying home and Whig pols being pounded at the polls. At the same time, Southern Democrats are adopting a still more radically proslavery line...and they were already incredibly proslavery. Lower South Whiggery collapses.

What's that do to the Whig Party? It had a bit of a northern tilt to begin with. It's just lost a lot of its voices in the Lower South which tend to be more compromise-minded toward the North as well as an important counterweight on the strong antislavery wing up in the North. The result is a party that's somewhat more antislavery (although still not primarily so) which makes the problem for Southern Whigs worse still. But there are enough conservative, proslavery Whigs in the North to maybe even some of that out. The thing is, they're all tainted by the fact that they voted for hated elements of the Compromise of 1850, particularly the Fugitive Slave Act and their leaders insist that that most odious law must be zealously enforced.

Maybe the Whigs could have recovered from that had the Kansas Question not opened when it did, but that kicks up in 1853 (a little) and then goes huge in 1854. With those conservative Whigs still around making the party look bad, actual antislavery Whigs have a sense that the party is not going to ever come around their way and that it may not be a longterm viable vehicle for them. The fallout from this is that the Whigs contest the presidential election in 1852, lose, and never do another. By the time 1856 rolls around, antislavery Whigs have gotten together with antislavery Dems (who are getting out of their own party for its more radical proslavery turn) to form the Republicans. Because it's a heterogeneous group of people united only on opposing slavery, the GOP is fundamentally an antislavery party first and everything else question mark. That makes it by default a Northern party, though there are some important players from the least enslaved slave states among the lot.

We can flip the script and do the same for the South, where the Democrats are now absolutely the Southern party and the question is how Southern they will be. Their dispute, internally, is over how far to go with slavery in the territories just as the GOP's uniting issue is that slavery will go no farther. These divisions harden steadily over the 1850s, particularly in the decade's back half when the Dems have two important crack-ups of their own. The first of these is over whether to accept the obviously fraudulent proslavery government of Kansas as a state (remember the proslavery guys cheated massively) or instead take on the antislavery Kansas government which is technically illegal and possibly treasonous but actually has the solid support of a majority of white Kansans. This is the point where the big names lay into each other, with James Buchanan on one side going his all for slavery as he has done for an entire long and miserable career and Stephen Douglas (him again) insisting that none of what shook out in Kansas was what he intended when he started the whole mess, the antislavery guys started it, but in the end you gotta give it to their government.

I'm a genealogist, so I want to respond from that perspective. Genealogists (and regular historians, too) routinely refer to the decennial and state censuses to learn about individuals. Your ancestor's age on the 1860 census may not have been something he or any member of his family reported.

In 1860, a census worker would have gone door to door to enumerate people. If no adult was available in your ancestor's home when the census worker stopped by, the information may have come from a neighbor, an in-law, a child, or a random person driving by in a cart. Extended family often lived in the same area and would give information about each other to the census worker. Cousin Willie's wife's mother might not actually know your ancestor's correct age even though she'd see him regularly. An estimate served the census taker's purpose.

Generally, we see age differences from census to census that are wrong within a couple of years. Sometimes there is as much as a decade's difference. These discrepancies are due to the varying dates the census was taken during the year, the unreliability of the informant, and human error.

Census workers made copying mistakes, misheard information, messed up their notes, and guessed at information on occasion. That's why we see garbled spellings of people's names - especially names that would have been considered unusual in that place at that time. Fewer people were literate in 1860, too.

Here are some resources that explain the census process and remark on the prevalence of errors:

"When the Census Taker Gets it Wrong," The FamilySearch Blog

"Five Common Genealogical Mistakes to Avoid When Using Census Records," The In-Depth Genealogist

"Finding Ancestors Hidden in the Census", Family Tree Magazine

And this Stack Exchange answer has a very good description of the process used to take the census in the mid-19th century.

As u/chocolatepot explained, the United States had never conscripted soldiers before the Civil War, so no one in 1860 would have been thinking about a draft or trying to figure out how to avoid serving because of that fear. The mistaken age on the census has a lot of alternate explanations, and your ancestor may not even have had a hand in creating the error.

Edit: I'm trying to find an article I think I read a few months ago in Prologue Magazine - the magazine of the National Archives - that addresses this, but I'm having no luck so far. If I find it I'll update this answer.

Edit 2: Additional articles:

FamilySearch Wiki: "Analyzing Census Data"

Census errors are so well-known in our field that I'm having trouble finding much - especially in the way of academic-level quality research and writing - that warns about them. Most of the discussion about census errors is how to combine the census with other evidence to arrive at a conclusion about a given fact such as a person's age. The literature universally seems to assume that researchers already know that the census isn't precise. I apologize to the mods if my answer isn't up to sub standards.

There was increasing fear of civil war for a long time in the leadup to the outbreak of fighting in the spring of 1861. It had some levels of recognition throughout the 1850s as unstable legislative compromises on slavery became law, such as the Fugitive Slave Act and the Kansas Nebraska Act. Abolitionists and their sympathizers heavily critiqued these pieces of legislation as handing over more state power to the slave aristocrats of the South. Abraham Lincoln made a name for himself nationwide as an anti-slavery republican in 1858 during his race for US Senate by heavily attacking pieces of legislation like this, which he contended would only serve the growth and entrenchment of slave aristocracy. With the advent of Bleeding Kansas and later, John Brown's infamous raid on Harper's Ferry, Virginia, more mainstream public attitudes began to lean on the idea that civil war could be on the horizon.

Talk of this exploded during the 1860 election season. 4 years earlier, the nascent Republican Party came in a solid second under a platform that called for banning slavery's territorial expansion and decried the Pierce administration as being conciliatory to slave supremacy in America. In 1858, the Republicans won the House for the very first time, which seemed to show a move towards anti-slavery in the Northern states. An effete and weak Democratic administration under Buchanan made a Republican victory in 1860 look likely, and to many, the Republicans began to seem like a radical political force. This fear was seemingly vindicated by many Southerners when John Brown attacked Harper's Ferry in the Fall of 1859, just a year before the election. After Brown was executed, Lincoln gave a speech in Kansas that would be derided by many slavery advocates as radical or dismissive, saying that Brown had agreed with them in thinking slavery wrong, but that cannot excuse his violence totally.

At the end of 1859, talk of political instability over slavery and the possibility of secession and civil war became much more widespread. Several cities began to host "union-saving" meetings, where citizens could gather to converse on the issues of slave power, abolition, and possible compromise. H.J Raymond of the New York Times, an ardent Republican paper at the time, spoke at one such meeting. He dismissed President Buchanan's idea that the talk of disunion then present was simply the political bluster that unfolds in an election season, and instead insisted that Southern states were beginning to craft a secession plan. It was also known at this time that Southern states were beginning to further fund and strengthen their state militia forces, and it was conjectured that the reasoning for this was to build an army to enforce secession. At the particular meeting Raymond was present at, there was talk of the possibility of southern secession. The idea of war was raised as a possibility, as well as that of an immediate negotiated settlement and recognition. Raymond rebuked the idea that any Northern party would interfere by force to end slavery entirely in the South.

Many papers used letterheads like "NATIONAL CRISIS" when talking about the continued establishment of unionist meetings in northern states and the building of strong army militias in southern states. Unionist meetings in Northern cities were largely grounds for Republicans/abolitionists entirely thought of as radical by conservative Southern Democrats, and their growth in 1859-60 seemed to indicate that abolitionists were radicalizing the North's civil society. Naval Commodore and former Senator Robert Stockton publicly refused to attend one such meeting in January 1860, stating that the meetings were "fallacious and calculated to deceive". Com. Stockton voiced his belief for sacrifice and concessions to be made by both Northern and Southern actors, and warned that "civil war will be the inevitable result" if this did not happen. This was over a year before the inauguration of Lincoln and the attack on Fort Sumter. That same month, Sen. William Rives spoke of the national crisis and the imminent danger of secession following the Harper's Ferry aftermath. He would eventually endorse secession and sit in the Confederate legislature. The secession debate and the merits of civil war were everpresent in the media and political spheres by the end of 1859.

Following Lincoln's nomination as Republican candidate that summer, the secession debate exploded. The Democratic Party was split over the approach to slavery. The Southern Democrats could not compromise on the Northern Democrats unwillingness to protect slave expansion, and nominated former VP Breckinridge, an advocate of slave expansion and a conditional supporter of secession. Breckinridge urged the press that Lincoln's election would represent civil chaos and upheaval for the Union. One paper stated that "after Lincoln is elected, some Southern communities... will refuse to let a postmaster appointed under his administration take office... US authorities will be interposed to enforce the law, then they will either be shot down or they will shoot somebody down. Then the people of the community will rise up against the United States government, and will be sustained by neighboring communities, until civil war with all its horrible butcheries envelops the land in a shroud of blood and carnage."

Lincoln was already known to be a fervent opponent of slavery, and had risen to prominence over the past 2 years as a Republican firebrand. Many recently established secessionist newspapers claimed Lincoln to be "softer on the niggers than Seward (his primary Republican opponent)" and that a Republican victory would bring misery and chaos to the nation.

After Lincoln's victory, the South went afire with civil protest. Effigies of Lincoln were burned in cities, scathing articles were printed attacking him, and civil disobedience became normalized. Southern states immediately began to call for state conventions to consider secession, with South Carolina becoming the first to leave the Union, mere weeks after Lincoln's victory. The state assembly had already issued a "resolution to call Abraham Lincoln's election as President a hostile act". With South Carolina gone, every other southern state followed suit.

This is only a brief image of the political crisis of 1859-60, but to put it succinctly; the country was aware that a political crisis over slavery was unfolding quickly. Individuals and organizations knew civil war was a possibility, and some even considered it inevitable. An entire secessionist civil infrastructure developed in the leadup to the election of 1860, and was well put together by the time Lincoln took office.

With the victory of a slavery opponent like Lincoln, and the immediate political backlash playing out in the South, civil war was almost totally agreed upon as inevitable by the end of 1860 and beginning of 1861. Many however believed that this would be over very quickly though, with the secession debate being solved in mere months. Few believed it would be so large and devastating and long as it turned out to be.

tl;dr: Yes, the average American in 1860 would likely have seen civil war as a strong possibility or inevitability. The election of 1860 was also largely seen as a referendum on the continued state of the Union, with none of the four options looking particularly able to defuse a situation that was already at its boiling point.