If you bought a car. Can you take it out of the lot driving it the same day? Do they give you like a special permit for driving it?

Yes you could drive it the same day and depending on models and years but you have all the paperwork that proves that you bought it so you could drive it

When you are ready to check-out and complete your transaction, go to the cashiers window and pay the remainder of your balance. You will be asked to sign all necessary DMV forms. KPA will process all title work for you, and you will be issued a temporary registration card, or you can elect to handle it yourself directly with a State DMV office. When you are ready to go you will be given the keys to your new vehicle and a copy of a “Gate Pass” to present to KPA security staff at the exit gate.