West LA, Palms, Venice... all that stuff is affordable, and not even CLOSE to the sketchy east LA scene. I've spent time in Mid-Wilshire, K-Town, and know people who live there for $600-800 even (with roommates though). I might have a different definition of bikable than others though-- I think anything within a half hour on bike (about 6-7 miles) is a reasonable bike commute. If you're looking to stick to that, or well under, probably West LA or Palms is the best bet.
Having a roommate (i.e., sharing a 2 BD with another person, not having someone in the same room as you!) would certainly be cheaper than getting a studio or a 1 BD, which in those same areas can range from ~$700-$1500.
If you're willing to drive, you can find SUPER CHEAP housing (I'm talking under $800 for rent and all utilities), but then you'd have to commute daily from the San Fernando Valley or something and deal with the miserable rush hour traffic on the 405.
Not going to lie, UCLA is pretty high up on my radar, despite the relatively low stipend (proximity to my family is nice). If I go, I plan to get my own place off-campus, probably in the West LA area. The current UCLA students are living there, on that same average (or lower) stipend, and they manage somehow, right?