Is it more proper for the questioner to travel with her mother as a mahram (if she is able to pass her exams), or is it more proper to avoid attending her cousin's wedding, which involves impermissible things, in order to avoid family problems?

It is impermissible for a woman to travel without a mahram (definition) (unmarriageable kin). If you are able to delay your mother's travel until after your exams so that you can accompany her, then this is what is obligatory. If that is not possible, and you believe that traveling with her will not harm your studies, then strive to accompany her, for Allah will reward you immensely. However, if you strongly suspect that it will harm your studies, then it is not incumbent upon you. Rather, strive to minimize the harm by ensuring that your mother has trustworthy female companions.

It is impermissible to attend a wedding that includes forbidden acts. You should offer advice to your relatives and congratulate them without being present at the forbidden acts. If it is possible to schedule the travel after the wedding to avoid attending it, then this is safer. If the wedding celebrations are free from forbidden acts, or if it is possible to change them, then attendance is permissible; in fact, it becomes obligatory if it is possible to remove the forbidden act. However, if there are forbidden acts that cannot be denounced, then attendance is forbidden, based on the Almighty's saying: "And leave those who take their religion as amusement and diversion" and "And among the people is he who buys amusing discourse to mislead [others] from the way of Allah." Strive for what is best and prioritize your mother's welfare.