Hmmm. Late Republic is tricky, but I'll try my best:
1) It didn't happen in real history, but technically it would have been possible, though every political reality militated against it.
The republican Roman army didn't have a career structure the way the Imperial army did. You joined, either as a volunteer or draftee, a legion raised for a campaign, and your rank would be assigned based on experience in previous campaigns, skills, and favouritism. ONce in the field, vacancies would open up as men are killed or wounded, invalided out or sent home, and these would be filled by the commanding officers as they saw fit, so a soldier could rise to standard bearer, optio or centurio or get a technical specialty. But at the end of the war, the legion was disbanded and he didmn't keep his rank. Come the next campaign, he might well be able to parlay it into new rank, or he might not. Of course the warlords of the Late Republic kept their cadres of experienced centurions for reuse, so this could be your opening. Both Caesar and Augustus (and IIRC Sulla and Pompey) rewarded their officer cadre with money and rank, including entry to the senate. That means that a successful centurion could, in theory, end up a Senator through favouritism.
The high command positions of the Roman army well into the second century were the exclusive preserve of the aristocracy, particularly the senatorial class. There is no career path from legionary to centurion to legate - legate is a political office. Again, in theory an upwardly mobile centurion could bedcome a military triburene, a legate, and eventually a praetor and propraetor or consul and then proconsul, or even receive an extraordinary command. But the political class at the time was difficult to manoeuvre even for second-generation arrivistes, let alone hapless superannuated warriors. I would not rate his chances high if he tried.
2) In theory, a freeman wasn't supposed to even serve in the army at all - the servile stigma stuck. Augustus probably created specific units for freedmen, the cohortes civium Romanorum, to keep them out of the legions. He probably didn't do this for no reason or out of personal prejudice. Of course the Late Republic also saw great manpower shortages, and we know that some of the warlords (namely Gnaeus Pompey) freed slaves expressly for combat duty. So it would be conceivable for a freedman to become a soldiers, perhaps even in the legions. Beyond that, what opened doors in the Late Republican army (and society) was patronage. If a noble commander was willing to stick his head out far enough to make a freedman centurion, then he could be centurion. He copuld not be senator, hence not rise to high command (except, in theory at least, through exceptional force of patronage - to a large extent the law was what the men with swords said it was).
Realistically, I'm guessing the best a freedman could hope for is honourable retirement as a navy or army noncom. the navy seems to have been more open even in Republican times.
3) In law, no. All army positions were strictly temporary, and after the army or unit dissolved you no longer held a rank. In practice, to an extent yes. Sons of proven veterans were probably given preferential opportunities, and the offspring of senatorial aristocrats would inherit their father's clientele of veterans and influence, allowing them to compete for high commands. In the Principate era, connections were very important at every level in the military, and there's little reason to assume it wasn't the case in the Republic. But you could not simply take over your father's centurionate - your father being a centurion just helped on the way to get there yourself.
4) If you were exceptionally rich, well connected and of proper birth you could (for want of a better word) buy a military command. Influence-peddling could secure the votes, political support the favour of the powerful, and depending on your aims you could find yourself provincial governor, promagistrate in charge of a war, or just senior officer in someone's military retinue. Positions in the legions were probably also tobe had in exchange for bribes, but this would hardly be the preserve of the exceptionally rich. Very wealthy provincials could also raise their own military units and put them at the disposal of a Roman general under their own or their friends' commands, but the border between corruption, personal wealth, organised crime and government was never as fluid as just then, so this may as well be regarded as an instance of alliance duty. What you could never do is formally purchase a commission because there were none.
5) Depending on your definition of the word 'humble'. Marius was a provincial aristocrat come to Rome and made very good indeed, but he was humble only in comparison to the senatorial families he clashed with. Sulla, Pompey, Caesar, Crassus and Mark Anthony all came from 'proper' backgrounds. I could not think of any true Republican bigwig who wasn't to the manner born, though some more so than others. You don't get careers from footslogger to emperor until the third century.
I'm hoping this helps a bit
Der Kessel ist voll Bärks!
Volker Bach
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