Is it permissible to cheat in exams to succeed in a mixed-gender study program with the intention of getting rid of it? And is it permissible to aid in that? If cheating is not permissible, is one sinful if they do not report someone who cheats, out of fear of causing family disputes?

Cheating in exams and elsewhere is forbidden by consensus, based on the generality of the Prophet's (peace and blessings be upon him) saying: "Whoever cheats is not of me." It is not permissible to cooperate in it. If you see someone cheating and do not participate in their cheating, then it is obligatory to prevent them or inform the supervisor if you are able. Otherwise, there is no blame upon you.