To some people it sounds like a stupid question but I was also taught that no serious question is stupid. What's common since to a
CJ student or LEO is not common sense to anyone else.
The only way you can legally carry a real badge is if you are a campus
security officer, and even though the laws vary state to state, unless you are a sworn LEO odds are you can't even bring a gun on campus. Most community
colleges have
security. Mine had a few
security officers and one sworn LEO that carried a weapon. Now when you move towards university level, the one I went to had a police department. Even though the were basically campus
security, they were sworn State LEOs who went to a training academy, carried weapons, and had full arrest authority.
As far as the felonies affecting the right to carry, it depends on what the offenses were. Best way to find out about that answer is to contact the local sheriff's office or state police.