This is fairly accurate. The southern slave-owning states had absolutely been worried about slavery being abolished. They had long been pushing for more protections of slavery, and trying to persuade the new territories to hold slaves as well. They were keenly interested in making sure that there were at least as many slave-holding states as non-slave states. There was even talk of invading the Caribbean, and setting up a series of new, slave owning states.

The 1860 election really hammered home this imbalance. Lincoln won without the electoral votes of a single southern state. This meant that the south, even as a united block, could not win majority elections in the federal government.

Now Lincoln had publicly stated that he was not going to outlaw slavery. He disliked the institution of slavery, but he felt very strongly that the federal government did not have the power to ban the practice. The southern states though, knew of his anti-slavery feelings, and with the practice becoming fairly unpopular internationally, feared that he would go back on his word and attempt to legislate against them.

As far as the Southern governments not representing their people, South Carolina is the most obvious example. The state legislature and governor positions didn't actually involve popular election until 1868. Ever since 1790, the state government of South Carolina declared that the legislators would appoint not only the state governor, but also all other legislative positions. Obviously, as a result, the state government was completely dominated by the plantation class who essentially appointed each other and took turns at various positions.

South Carolina also had the most to fear from Lincoln. With a population which composed more slaves than free men, the federal government and John Brown style abolitionists drove the legislature into a rather paranoid frenzy.

Was secession popularly supported? Yes and no. You had voices on all sides of the spectrum. McPherson certainly notes that letters written during the lead up to war written by plantation-owners contained roughly twice as much "Nationalist" language than poorer whites. Newspapers supported both sides of the argument. There may have been plenty of white southerners who weren't as invested economically in slavery, but damned if they were going to let the yankees criticize them. Propaganda followed the usual drum-beating approach. The North is going to impede on our freedom, and you're less of a man if you don't fight against that!

Ultimately though, I do feel that the Confederate governments led their populations into war hastily. Once shots had been fired though, the Southern population were certainly willing to fight as a matter of pride.

Sources:

Edward Ayers In the Presence of Mine Enemies, James McPherson's For Cause and Comrades, and McPherson and William Cooper's Writing the Civil War.

Stephanie McCurry's Confederate Reckoning covers this question perfectly. Her book is concerned with the role of people who didn't hold power in the Confederacy--ie, non-slaveholding whites, women, and blacks. More than anything, we should remember that women and blacks were people who were not allowed to vote in an election that would determine their future as much as anyone else's.

One of the first reminders that McCurry makes is that the Confederacy was conceived illiberally:

What secessionists set out to build was something entirely new in the history of nations: a modern, proslavery and antidemocratic state, dedicated to the proposition that all men are not created equal. Confederates were fully caught up in the turbulent currents of history that roiled the hemisphere in the age of emancipation; their proslavery experiment was part of a far larger struggle then being waged over slavery, democracy, and the powers of nation-states. Theirs was a nation founded in defiance of the spirit of the age.

Let's start with South Carolina. It's is always a weird example when it comes to secession because it is the state with the largest fears of a race war like that in Haiti. It was also a single party state; South Carolina's voters didn't get to vote on their governor, US senators, state senators, many local officials, and electoral college (ie, the electorate didn't get to vote for president in 1860; the legislature did that for them). There, voters were allowed to go "to the polls in a climate of political terror, surrounded by armed companies of men" in order to vote in a show election for the state legislature in tickets that gave "voters" the right to support single party rule in the state. Consequently, nearly everyone elected to the state legislature had publicly advocated for immediate secession from the Union, and this was the closest South Carolina's voters got to voting for secession.

This chapter goes on to show how other states (or rather, slaveholders governing them) avoided letting the general public vote for secession because they didn't want to risk losing those elections.

Georgia and Texas, as in the /r/history thread you mention, were on the other side of the spectrum. Sam Huston was completely against secession and did everything he could to avoid it. Even then, the convention in Texas voted 166 for and 8 against secession, but this had to be confirmed by a popular vote. States that required a popular vote were much less likely to secede than South Carolina, for example. Georgia's vote was incredibly close, but the governor fudged the election results in order to secede with the rest of the lower South.

So yes, I would say that the democratic process then was a lot different from how we would expect it to work now.

The state legislatures were indeed representing the will of the majority of the white population in secession. Southern politicians played a major role in creating this support but by the time they acted support for secession was widespread among all classes of white population in the south.

Southern whites genuinely believed that abolition of all slavery would be passed within a short amount of time. They believed that abolition was contrary to god's will as he had expressly allowed it in the bible. They believed that abolition would destroy the southern economy and plunge everybody into poverty. Perhaps most importantly they feared that without the social order of slavery blacks would become idle, then criminal and finally would engage in rebellion and murder against whites.

To start take a look at the relevant results of the 1860 census. Slave owning elites may be a bit of a stretch. One third of all southern families owned a slave. In South Carolina arguably where secession has its roots 46% of all households owned slaves. 46% is hardly just a slave owning elite.

You also have to consider that not all slaves worked in the fields. If you went to the grocery a slave would likely carry your purchases home for you. If you went to a party there would be slaves to take your coat, stable your horses and unseen preparing much of the food. Slaves cleaned the streets, cared for children, worked in bakeries, and in general did all the undesirable tasks. I'm not saying Southern Whites were lazy rather that the presence of a controlled group of persons allowed them to avoid some tasks. Abolition didn't just affect the elites it changed the entire southern way of life.

Returning to abolition and god's will. The major churches in the United States fractured over the issue of slavery well before the Civil War. Presbyterian in 1837, Methodist in 1844 and Baptist in 1845. In the years leading up to the war many southern congregations discussed how the North was trying to disrupt God's natural order by freeing blacks. The language interpreting god's will into politics increased in the years just before the civil war in both the North and the South. As each group insisted more on God's favor Religious language became increasingly common from politicians such as Abraham Lincoln and Jefferson Davis neither of which showed a particularly religious nature earlier in their life.

As for my other claims they can be read in pamphlets that circulated throughout the South prior to and during the Civil War. While true that these pamphlets were mostly published by wealthy slave owners this was in large part due to this group having the money to publish such things. The spirit in Southern media both in newspapers and pamphlets was very much a we are all in this together attitude. Freeing the slaves meant the end of the Southern way of life for all whites slave owner or not. To better understand the rational of the South read John Townsend The Doom of Slavery in the Union: Its safety out of it

By the time the first shots were fired there was popular support from far more than just a few "slave-owning elites." The impetus to drum up this support did come from the wealthy slave owners they couldn't have manufactured the support from thin air. The ideas were already present in the southern population the politicians turned the ideas into a cause.

The short answer, yes. The longer answer...

On secession, legislatures did not make the decision to leave the Union, there was a slightly different logic involved. The US Constitution, the big one that created the federal government was debated, written, and signed by a group of men who represented individual people. It was meant to be something entirely distinct from any national legislature under the Articles of Confederation. Thus the line in the federal constitution "We the People." They aren't kidding. They really mean, all of the people who will be governed by the document (there are some logical turns in republican/democratic/constitutional theory that help them to account for people who did not support ratification.) It was done as something completely distinct from the sitting national legislature.

Fast forward to 1860. Slave-holding southerners are fed up with the federal government and worried that it will threaten slavery. They want to leave the Union, so how do they do it? The same way that the union was formed in the first place. The individual states called constitutional conventions and the men who represented the people writ large are the ones who voted to leave the Union. If you look at the ordinances of secession, each and every one of them says something like "the people in convention assembled." It's not fancy language or just a nice turn of phrase. They mean, representatives of the people who are governed are the ones who consented to the creation of the federal government, they are the ones who are officially withdrawing that consent and seceding from the Union.

Was secession popularly supported? The evidence that I can find suggests yes. Texas was the only southern state to submit their ordinance of secession to a popular vote before adopting it. 3/4 of the voting public voted in favor of leaving the Union, suggesting that yes. There was fairly widespread popular support for secession (though not universal support, there were some unionists running around in the south).

Sources: Ordinances of secession For the Texas fact see: Richard White, Law in American History vol 1, pg 386. White probably also has the bits about the people in convention assembled, but I'm sure there is a better secondary source for it. If anyone really wants it, let me know and I'll dig it up. For primary sources, you can read the Federalist Papers for the national constitution and then compare the logic there with writing from John C. Calhoun.

edited to clarify a sentence and formatting because some days there just isn't enough coffee.

From Eric Foner, I understood that yeoman and poor white farmers (white free labor) did not necessarily share a natural sympathy with the slave holding elite. I am hoping someone more knowledgeable than me would comment on this perspective vis-a-vis OP's question.