If your in an elevator accelerating downwards faster than gravity, would you be pushed up to the ceiling? Since gravity cannot match the acceleration of the elevator, you're wouldn't be able to move at a constant velocity relative to the elevator.
Yes. Although the elevator car wouldn't be able to fall faster than gravity - you would need to have a cable pulling it down.
One of the most important principles of physics is that you can't tell if you are in a gravity field or in an accelerating box.