1. Trust your best friend of 11 years
2. As for what happened, well there's no way for you to know exactly what happened if indeed anything did happen, so
3. If you really decide to keep seeing him outside the home, find out as much as you can about him, start with your best friend and her partner (his bf I think you said?)
Ask for specifics about past supposed thefts etc. think about his personality, his life, career history etc. is there anything that doesn't fit? Has he been in stable employment for a while etc etc that sort of stuff.
It does seem very unusual that someone who's just been accused of stealing pulls out $100 and gives it to the accuser. It seems like guilt, as opposed to high morals.
But trust your BF of 11 years - you've been friends for 11 years for a reason and if it did so happen to be some crazy set-up then she was lied to as well by the sister.