I’m going to answer this as best I can remember, and from what I know of the undergrad process, which is mostly second-hand info from my ex-wife and friends at Oxford.
Yes and no. Typically an undergrad selects three colleges as his or her preference. Each college is free to accept or reject the student’s application, so while a student is free to pick any college he or she wants, in the end it’s up to the colleges as to whether they’ll take that student or not. I didn’t know of anyone who got rejected by all three of their choices, but it happens. Then you get assigned to a college which didn’t have enough students declaring a preference to them.
What it essentially means is the student has to take a bit of a blind gamble. Obviously if you’re one of the top candidates, you’ll probably get your first pick. Every year there’s a “league table” among the colleges for exam results, and the colleges want to take the best students available to get to the top of the league (ETA: Norrington Table, that’s what it’s called). But if you’re in the middle of the pack…it might be better to not go for the more established colleges with more academic prowess, lest you apply for “Christ Church, Balliol, New” and have them all reject you, leaving you to be assigned to Green College or whatever.
Apart from what I’ve suggested above–students want to study in the more established, more academically-rated colleges–students pick colleges for the same reasons incoming freshmen in the US pick their universities. One college might be more known for business, another for PPE (philosophy, politics, and economics–aka the hack’s degree), still another for Classics. Perhaps someone wants to study with a specific professor who’s well known or who someone they’ve known has recommended. Some students go to the college their father or other relative has attended–most colleges have preferences for “legacy” students. In short, there are colleges for toffs, there are colleges for hacks, there are colleges for serious students, and some colleges are for all three.
There are other reasons to attend one college or another. The college I attended, Jesus College Oxford, is traditionally the college for students from Wales. It isn’t like the old days when close to 75% of its student body was Welsh, but the percentage is still fairly high. If you want to study Welsh or Celtic languages at Oxford, it’s your best bet; if you want to go into politics in Wales, it’s a good choice too. Other colleges have similar tendencies in their student body.
Not that I’m aware of. I’m thinking it’s fairly rare; then again, I didn’t know that many undergrads, although almost all of the ones I did know stayed at Jesus College for their three years (the ones who didn’t were generally dropouts).
Oxford is really pushing the idea that “an Oxford degree is an Oxford degree” now. But it’s a new concept, tied to the greater centralization of administration (which, in turn, is a result of student fees and the concentration of money in central administration). Most old members don’t even say that they’re Oxford grads: they’re Jesus College grads or Somerville grads or St. Catz (Catherine’s) grads. My gut feeling is that if you’re going into business or hard science going to one of the newer colleges isn’t a burden, and it may be a bonus. But there probably still is some discrimination towards the new colleges (as opposed to New College). That’s all relative, though: a grad from any Oxbridge college is going to be thought of more highly than even the top grads from a “redbrick” university like Leeds or Manchester (a shame as I attended Leeds too and in some respects you’ll certainly get a better education there).