Is it bad if you do push ups and sit ups every night?

Probably not, unless you do too many of them. You need to allow your muscles to heal in between workouts. Most people work their abs (sit ups) daily. Arms and chest are usually done every other day, but a good warm up or light workout every day shouldn't be a problem.

I don't see how. In basic training, I can't even begin to estimate how many push ups I did on a daily basis. Are we talking about 50 before you go to bed or multiple sets of multiple exercises??