What is the ruling on enrolling in martial arts courses that require bowing to the trainer and others, with the possibility of not doing so? Is it permissible to return a greeting by striking one fist against the other palm?

Bowing as a greeting is not permissible in Islamic law, whether in sports or otherwise. If bowing is optional, then there is no harm in joining courses, provided one apologizes for not bowing and replaces it with a handshake. As for striking the fist of one hand against the palm of the other, there is no objection to it as long as it is a custom, for customs are permissible as long as they do not contravene Islamic law.