Basically, No. But bad teeth can cause sinus problems, although that is relatively rare as well. A sinus ct can usually tell what is the source of the problems.
Not to my knowledge. Perhaps in very unusual cases it could happen if you have a severely disease maxillary sinus chronically and the roots of your teeth are bulging through the floor of the sinus. Tumors of the maxillary may affect the teeth if the tumor erodes. Sinusitis, however, often causes dental pain, without any erosion.