Commuting from Santa Monice to HQ every night would be awful. The closeness on the map can be deceiving when there's an ocean of cars in between.
I personally would probably not move right by HQ if I had a car and sufficient funds, but that's just me. Though, I don't think it's dangerous, really (if you can ignore the gang graffiti). Where you choose to go kind of depends on what you're into. Los Feliz, Echo Park, and Silver Lake are all kind of hip up-and-coming areas. West Hollywood is nice. There is a large gay community there, so you get what comes along with a lot of young guys with discretionary income. That is, it's clean, and there's lots of restaurants and bars, etc. Commuting from Burbank wouldn't be out of the question either. That's your suburban neighborhood kind of vibe. If you want a yard, you'll probably look in that direction.