Do they have them all year long or is there a "season" for car racing? / January to November typically.
What about the cars that get smashed up? / They turn into spare parts or get junked if severe damage happens. The expensive and most valuable pieces are the engine and frame. A lot of time and effort are put into these key components.
Do they let the race car driver use a different car for the next race or does he have to fix the car he has? / If he can fix the one he broke, that's OK. Usually every team has a backup car he can use as well.
I'm sure other folks will expand on answers to yur questions. Thanks for asking.
Welcome aboard!