Is it bad to store a prescription mouth guard in a mouth rinse every night after brushing it with toothpaste? Would it weaken the guard?

Cleaner. Rather than mouth wash use Denture Cleaner (Pollident, Efferdent, etc). Make sure to wear appliance EXACTLY as directed.
Yes, it is. Store your mouthguard in denture cleaner and do not use toothpaste on it!
Yes and yes. A prescription mouth guard is often supposed to be stored dry after being cleaned. As well, many times the dentist will provide you with written care instructions as well. Your technique sounds like it will damage the mouth guard, so double check with your dentist.
Yes. Follow the instructions that your dentist gave you. Mouth rinses contain alcohol and other chemicals that can degrade the mouth guard. Brushing with toothpaste can be abrasive and wear the mouth guard. Use plain toothbrush for brushing a denture cleaner or an ultrasonic cleaner. If you want to freshen the mouth guard you can rinse it. In mouthwash but. To soak it.