Restriction Up and Down movement.
You may try to disable the mouselook and bind all the movements to WASD with a gameui, but I suggest to take a look at those old tutorials that explain how to build a vehicle out of brushes. it has been done already and it's pretty easy to do.
This sounds very similar to the situation at the beginning of the original game: you're trapped in the moving container but free to move around inside it without getting stuck in the walls, as would happen if the player was actually inside the model.
What actually happens is that the player is in a separate room, and their position is continuously recorded relative to a reference point_viewproxy. Then the player viewpoint 'camera' is positioned with the same offset relative to a 'twin' point_viewproxy that moves with the container. (Listen to the commentary track for a more thorough explanation.)
This seems to me like the perfect thing for your car idea 
Other Portal 2 maps: Medusa Glare
Portal 1 maps: Try Anything Twice | Manic Mechanic
Is that why crouching has no effect in that scene until it's over?
[spoiler][SP] Alternate[/spoiler]