To answer all of these questions:
1.) The UCLA website explains its processes here (The "request items" entry is most important, but all sections have some information). In addition, this guide is important
2.) There's no stated limit, but the more you ask for, the longer preparations take (see previous links).
3.) Access is free.
Go for study copies, you can get a one day turnaround and have virtually no limit. If you ask for research copies, you have to wait two weeks and they will generally give you a hard limit (I usually got 2 or 4 when I asked). Once somebody has had a research copy transferred, it becomes a study copy.
When I am asked if I could watch one thing, I would get
The Game Show Show. Although it's by today's standards looks like Public Access, you get interviews with people who normally didn't do interviews (Ralph Andrews, Jack Barry), they're not puff pieces, and the transitions are simply pictures off the TV of shows of the era (1971).
As much as access is free, I always try to donate something.
If you don't want to go through the hassle, go to Paley. Access is pretty instantaneous, although you only get two hours.
--Mike