Runaway or "roaming" slaves were known in Ancient Rome-- there are specific terms for this. Runaways were called fugitivi (the source of our word "fugitive") and "roaming slaves" were errones; it appears that the distinction is that runaways were gone for good, while the errones are sometimes called "truants" -- they see to be folks who're not where they're supposed to, maybe drunk in town, but not permanently fled; but its not completely clear.

Romans tended to be pretty meticulous about documentation, and the moved in social "circles" of people who knew each other well. So Mr "New guy", showing up in a new town-- well, claiming to be some known individual wouldn't be smart, even at the other end of the Empire. Romans sent officials all over the empire, and if you were claiming to be a Tullius-- there was a fair chance some other Tullius might show up one day.

There's actually discussion during the Republic of the possibility of a law to make slaves wear distinctive clothing, but the Senate decides that would be a bad idea, as it would make the slaves aware of how numerous they were. So a slave wouldn't have stolen his owners' clothing-- that would likely have attracted suspicion; he'd have done better looking like a random urban worker of a lower class.

So it appears to be the case that Romans couldn't tell a slave on sight-- and an escaped slave would have done well to disappear somewhere where folks might not look for him. Taking his owner's name wouldn't seem smart, but an industrious soul in a new town could presumably have found employment and stayed hidden. He'd have had to avoid certain kinds of transactions which would call attention, Romans had an efficient tax and property title regime, so depending on just when and where he was, his life would have been circumscribed.

Remember also that Rome had many people who were not slaves, but also were not citizens. Our runaway slave could far more easily passed as "free" than as a "citizen".

Slave Naming in Republican Rome

SOME MATTERS FAVOURING THE RUNAWAY SLAVE HYPOTHESIS IN PHILEMON

THINKING SLAVE AND FREE IN COORDINATES

and for evidence of the unhappy fate of caught slaves, see:

Crucifixion as Spectacle in Roman Campania detailing slave crucifixions under the lex Puteolana

a) Can a slave try running away to a different town or city and living there?

Yes. There are recorded anecdotes of runaway slaves later being discovered living in Roman towns or cities.

b) And lie about being a citizen?

More difficult but certainly possible. Roman legal sources often show more concern about slaves running away to other masters than running away to pose as citizens, but it was possible to pretend to be a citizen, particularly in a large city, especially for some slaves. For others, e.g., slaves with prominent whipping scars which will be obvious as soon as they step into the public baths, maybe less likely.

Why is it more difficult? Roman society has been described as a face-to-face honor society. Social ties, ancestry, relationships, etc. are very important in establishing one's public identity and status. If a stranger shows up in, say, Pompeii claiming to be a local citizen, people will want to know why they've never heard of you or your family. If you say you're a citizen from somewhere else, say, Rome, people might want to know what kind of citizen of Rome you are, which entails who your relatives are, what prominent people you're connected to, etc. There's not necessarily a bunch of atomistic citizen-shaped holes in society waiting to be filled; people will be curious about your whole package of relationships because that's what makes up a real Roman citizen's identity. As a disconnected individual with no family, history, ties, etc., it's hard to fit unsuspiciously into citizen society, especially in smaller towns - it’s not the type of society that has a place for people without relationships to anybody. This may be why Roman law seemed to contemplate slaves running away to be slaves of other masters - that was the type of place that was easily available.

This sort of passive mechanism may have been somewhat easier to evade in big cities due to the sheer churn and turnover of people diluting the effect of the face-to-face society. Conversely, though, chance encounters with people specifically looking for you (or at least from people who would recognize you if they happened to see you) are more likely. There are always slave-catchers and informants and generally people who might recognize you circulating around in a big city, there are notices posted about runaways, and generally there are various mechanisms to make it a risky environment.

But difficult or not, it did happen. There are even occasional reports of slaves passing as free successfully enough that they (or their children) held public office.

c) Were there mechanisms to stop this and could those be bypassed?

Yes and yes, as partially discussed above. There were various public and private mechanisms to catch runaways or to identify incognito slaves. Masters - particularly wealthy masters of many slaves - often had networks of friends and contacts in various towns and cities, and they could and did ask these to be on the lookout for a runaway slave. Owners could also post notices and rewards, hire professional slavecatchers, who might do the same sort of canvassing, and call on government officials, who were required to assist in searches for fugitive slaves.

These public and private mechanisms for surveillance and slave-catching appear to have been effective enough that Romans assumed that most runaways would eventually be caught, even effective enough that we have some records of masters all but daring slaves to try it and see what happens, or taunting those who did run away and were caught. However, they were not of course completely foolproof and some slaves managed to get away for good, either to a new master or to actual freedom.

Measures to stop a slave who had tried to run away once from running away again might include chains, collars with tags naming the owner, or more harshly facial tattooing meant to forever mark the slave as such or even mutilation to make escape more difficult (such as cutting off a foot to prevent running). These likely had deterrent effects as well as the obvious effects, since obviously risking having bits of yourself cut off is a lot to risk, nor did anybody want to be forcibly facially tattooed with something that would probably permanently ruin their chances for good treatment, better positions in the slave hierarchy, etc.

d) What if some real citizens backed up a runaway slave's false claim of citizenship?

This sort of dispute is destined to become a legal case before a magistrate, and both sides can gather evidence and present witnesses and such. If the community maintained citizenship rolls, as Rome itself and likely many other communities did, these could be consulted. Documentation (such as the bronze military diplomas issued to auxiliaries granted citizenship) might be expected if the claimed citizenship is supposed to have been a recent grant; records, family ties, former patrons, etc. might be expected in case of claimed longstanding citizenship. This type of case is a subset of a broader class of cases adjudicating someone's status, which were nothing unusual for a magistrate.

There were harsh legal penalties for aiding runaway slaves, so you're going to have some work to do to convince a bunch of citizens to walk into court and testify for you. This might be another reason the Roman law contemplated slaves running away to other masters: offering to become their slave is an incentive for them to risk the law's punishment by helping you get away from your first, presumably worse master.