UCLA is located in Westwood which is part of the Westside.
North Hollywood is about 10 miles away from UCLA. Like Gomer said, traffic in that area is horrible. It doesn't matter if you take city streets or the freeway, it's bumper-to-bumper traffic. Also, many people are taking earlier schedules, so the traffic starts around 6:00am. North Hollywood is known as a gay community, just so you know. Some parts are rundown, some parts have great restaurants and shops. The Hollywood hills are where a bunch of movie stars live.
The area around the hospital is not affordable unless you take into account how much time you will spend in traffic if you don't live close. Most students live North of Wilshire and East of Sepulveda. The closer you are to campus, the more expensive. Two blocks can make a difference in rent. If you look South of Wilshire Blvd, things are a little bit more reasonable. UCLA also operates an amazing vanpool/commuter program if you should decide to live further away.
Sorry, I don't know anything about their New Grad programs.
I believe that UCSD (University of California, San Diego) Medical Center is a teaching hospital. It's located in La Jolla. The area is really nice. Traffic isn't as bad as Westwood.
Hope this helps.
Good luck on your decision!