Sup, fellow Bruin! Well, actually I'm not going here for grad school, but I have been here for almost 5 years.
I can't really speak for how accurate the crime data is, as I have never experienced crime in any capacity (with the exception of hobos in Westwood, which isn't so common but I am prone to attracting them).
The apartments just west of UCLA vary. I know that a lot of people say that it is loud, but I have literally lived in or spent time at five different apartments in the area. What I can tell you is that if you live near the frat houses, you're pretty much guaranteed to have a loud number of Thursday-Sundays during the year. But apartments along Veteran or a couple blocks north (around Gayley/Kelton for example) are actually really quiet. The ones along Veteran have a lovely view of a cemetary, so if you're into that, that's something. The area is generally mostly undergrads, though. I recall one of my friends split a studio on Kelton with someone else with all utilities included for $750 a month (per person), which sounds like it's just within your budget. It was right across from a party house though, so your mileage may vary.
If you really want something quiet, I could also suggest south of Wilshire (there are a bunch of apartments just south of Wilshire Blvd around Westwood Blvd). Their apartments seem to be higher quality for a lower price. With 4 roommates total, we were able to afford a 2 bedroom 2 bath at $600~ a month (per person), so that would mean I would have been able to score my own room at $1200~ a month. Plus, there's a shuttle (bruinbus) that goes from Wilshire/Gayley to campus and back (from ~7:30 AM - 7PM).
I currently live in palms, and the area is fine, and really convenient (grocery stores everywhere!). I take the bus, which is about half an hour there and half an hour back, not including waiting.
You should be able to find an apartment if you want to move in August/September. A lot of the time, if the lease has awkward start/end times, people sublet the apartments over the summer, so moving in early would not be a requirement.
Can't say a lick about grad housing, since I have never been a grad student here. Good luck!