A little info from my experience and what I've heard.

Registration - doesn't matter, I took my car and it's registered in my fathers name, they don't look at registration, just make sure you have a valid license.

Insurance - as far as I know, no company will cover an damage caused in an accident on the track, don't see them stopping coverage though just because you've taken your car to the track. On a side note, there's rumors about geico reps etc. going to tracks and checking vins then cancelling policies or some BS, that's only what I've heard online so take it with a grain of salt