My gym charges extra for ICE (a competitor of Spinning®). You can sign up for an 8-week program, I think it's two or three classes a week, which guarantees you a slot, otherwise it's first come-first served.

It probably varies a lot by gym.

I don't spin (I need shorter crankarms, the one thing on those bikes that isn't adjustable), so not much help here - but depending on the gym, the bikes and the amount of time between classes, you may be able to bring your own pedals and saddle/seatpost. Some gyms have SPD pedals on their bikes, so if that's what you use you could wear your own shoes.

Definitely shop around.