Would A Bad Thermostat Cause The Car To Over Heat And Or Not Want To Stay Running Upon Starting It?

Yes. The thermostat is not like the thermostat in your house. It's a spring-loaded valve that controls how much coolant flows through your engine. If it is stuck closed it can cause the car to overheat and to run rough. If it is stuck open it can cause the car to never get up to proper operating temperature and again to run rough.

The good news is it's a cheap part (a few bucks) and usually relatively quick to replace.

This is a situation of cause and affect. The heat of the engine causes the thermostat to change its reading, if the thermostat was broken it would not affect the engine's heat.

If possible if my thermostat is broken my heather core can't blow warm air on my car