1. Nobody's providing a car for you to drive, instructors are just along for the ride. If you're a student, yes, the passenger (your instructor) needs the same level of safety equipment as the driver.

2. You can enter any car that you have permission to drive on a race track. (I can neither confirm nor deny that people sometimes use rental cars) While the level of risk of an HPDE event is considerably smaller than a race, it is also considerably greater than something like an autocross... the general "don't put anything on a race track that you aren't willing to walk away from after you crash it into a wall" rule applies.