1. Is it permissible for my wife to go to university alone?

If the university's distance is considered a journey, then it is not permissible for the wife to go to it alone except with a mahram, due to the Prophet's (peace be upon him) saying: "A woman should not travel except with a mahram." However, if it is close and not considered a journey, and there is no fear of harm befalling her, and her going out does not result in khalwa (private seclusion) with a non-mahram man, then there is no harm in her going.