Cheers to lloannna who gave you excellent answers. Everything is pretty much covered. Downtown LA is not too far from LAX, but it depends on the time of your flight and you need to consider the traffic conditions at that time.

I’d stay overnight in Anaheim and then decide:

—if the second night is a Friday or Saturday, or it’s between May and September, and I wanted to watch the fireworks and/or Fantasmic, I’d stay the second night in Anaheim, BUT
—if the second night is a night other than Friday or Saturday, and it’s between September and May, I’d know that the park closes at 8pm and there are no fireworks, and I’d spend the second and third nights in Hollywood.

In a car in the dead of night it can take as little as 50 minutes to get from Disneyland to Universal City, excluding time to get to your car and get out of the Mickey and Friends parking structure, and also excluding the time it takes to find a spot at Universal, park, and walk to the entrance. Any shuttle or taxi will take more time than that (probably more like 75 minutes,) mostly because there’s more traffic at non-3am times—but you get to skip the parking hassles.

I’ve never been an adult tourist from out of state at Disneyland, so I’m not familiar with the details of most transit operations other than the bus and metro, which will take a VERY big chunk out of your three days. I’ve misplaced my list of Disney email addresses, however, Guest Relations is always happy to talk to you:

(714) 781-4565

These people run a shuttle from Anaheim to Universal. It’s more than I’d pay, but since you don’t want to drive, it’s probably the fastest alternative. The timing won’t seriously cut into anything you’d want to do at Universal Studios (they open later in the day than Disneyland does) and there’s no law that says you have to take the shuttle back to Anaheim. You’d have to get yourself to one of the hotels in the Anaheim area that they pick up from, but the city has set up a little bus route to take people around the various hotels, the convention center, and the Disneyland Resort, so that shouldn’t be hard. The tour site says their shuttle takes 90 minutes: that probably includes some wandering around or “okay, everyone, sit down so we can get moving” pad time, since they’re operating outside normal rush hours.

http://www.allanaheimtours.com/Body.asp?tour=ANA-B0041&page=TourDetails

I suspect, however, that the Fullerton hostel (that’s the closest one to Disneyland) people probably know this stuff far better than I do, as they’ve certainly had occasion to research it for someone before. Their email address is:

hifull@aol.com

I’ve never tried them in particular, but most hostel staffs are really nice about answering questions.

I see no good reason to stay overnight in downtown LA—Hollywood is a lot more interesting after 5pm (though again, don’t go wandering around the side streets alone at night). I lived in LA for most of my childhood plus three full years as a working adult and I’ve never had occasion to do anything downtown at night (and when I’ve had to drive through downtown after dark, usually because of a freeway closure, it’s been eerily quiet and under-lit.) Amongst other things, I think it’d be quite a challenge to find a place to eat dinner downtown (especially as compared with Hollywood and West Hollywood.) The metro system runs pretty late into the night if there really is something you want to do downtown after dark, and it starts up early in the morning, and so I can’t think of a good reason to check out of or even just walk by a good hostel in Hollywood in order to sleep downtown one night.

(Los Angeles is a fascinating place, architecturally speaking, and the area is home to some great museums: my favorites are the museum of Natural History and the Museum of Tolerance—there’s also a great museum of Japanese-American heritage, and of course, Griffith Park and Observatory isn’t too far away. It’s all 10am-5pm stuff, though, unless you’re seeing a show, and if you are seeing a show, you’re likely to be in Hollywood.)