Is marriage fate or choice? And is my brother's completion of the marriage – despite his discomfort, the girl's non-adherence to religious principles, and his falling into quarrels with his mother and sisters because of her – considered an act of disobedience to his mother?

Everything in the universe is by Allah's decree, but the servant has a choice, and his will is subservient to Allah's will. The brother must obey his mother and refrain from marrying a girl his mother is not pleased with, because obeying one's mother is an obligation (fard), while marrying a specific girl is not. The engagement is merely a promise and can be annulled when necessary. The brother should be cautious about causing problems with his mother and sisters to avoid filial disobedience (ʿuqūq al-wālidayn) and cutting off ties of kinship (qaṭīʿat al-raḥim), both of which are major sins.