Is the designated prayer area at the college considered a mosque, knowing that Fajr and Jumu'ah prayers are not performed there, but the remaining prayers are performed there in congregation (by both young men and women), and it is referred to as a mosque? And based on this, is it permissible for a menstruating woman to stay there?

The prayer areas in workplaces do not take the full ruling of a mosque. A menstruating woman is permitted to enter them. The difference between them and a mosque, as stated by Sheikh Al-Uthaymeen, may Allah have mercy on him, is that a mosque is an endowment and cannot be sold, whereas a prayer area can be abandoned or sold depending on the place where it is located.