It may look seedy south of wilshire, but anywhere on the westside is pretty safe. It's when you get to Venice or Culver City (another option for UCLA students and the Culver City Bus is another great local bus that has a couple of lines that go up to UCLA) that some spots get a little iffy. You just have to be smart. I've always felt that the problem with LA is that each area feels like it's own little community and you sorta forget that you're living in a big city.
Some parts of culver city are on the up and ups. The downtown area is quite nice actually and I think the 3 line of the culver city bus goes up to UCLA.
And as for timing, I think giving yourself 2 days before the move in date is not the best idea. YOu have to remember that there are only so many apts and right around the time you are moving out, about 10,000 other students are as well. I always looked for places about 2 months before I needed to move in. In Westwood I gave myself more time. I signed a lease for a place I was renting in August in April of the same year. After May it became very difficult to find housing in Westwood. In places like Culver City and Palms, I doubt you need that much time.