I have severe claustrophobia and heard that the parking structure is underground. Can I be dropped off at the tram pick-up and wait for my driver there after he parks the car? Are there stair options instead of elevators,?

Yes, I used to work there. You can even walk up the hill. Just notify the security guard usually stationed at the bottom of the hill.

This is a very spacious and sprawled museum and you can avoid the parking or the elevators.

You can avoid the parking structure and directly go to the team. The museum has stairs for every floor too. It's quite spacious.

Yes you don't have to go into the parking lot, just be dropped at the entrance and climb the stairs to the tramp