What is entailed for someone who vows not to miss a prayer or be late for it, and then is late for, makes up, or misses obligatory prayers? And does the ruling differ if the vow is perpetual?

This vow of yours is not considered valid by the majority of scholars, and you are not obligated to fulfill it. This is because it is an commitment to something already obligatory by the original tenets of Islamic law; a duty that is already obligatory is not affected by a vow.