It's just polite to wait. I would never ever just drive off until I knew she was in the car and it started. I would do the same if it was a guy too. If it's late, I'd hate to strand someone if their car didn't start.
Same with dropping someone off at their home. It's just polite to make sure they get inside, didn't forget their keys or whatever. Then once they are inside it's polite for them to wave or flash the outside light something to show that everything is OK.
it depends on if he adheres to chivalry. in this day and age, with cell phones, it's not nearly as big a deal. if the girl shows up in a car that is barely running, he's more apt to. if she shows up in a brand new car, not really necessary. some girls will sabotage their cars or say it won't work to be invited home by a guy. if the guy cares about the girl, yes he should see her to her car, and wait there until she's started out. it shows her that he cares. if he doesn't even walk her to her car, it's a sign that he's not that into her.