What problems if any will I have if I buy a right handed AR15 and shoot it left handed?
None. The AR is one of the most friendly rifles out there for lefties.
The charging handle is ambidextrous and the shell deflector keeps brass out of your face.
The mag release is easy to operate with your thumb when grabbing an empty, and the safety can be operated with your forefinger (or you can easily install an ambi safety).