an applicant’s religious beliefs? Are they allowed to advertise as such, even requiring an applicant to prove religious affiliation? If there is a law, what is it exactly? If not, how do they get away with it? Is there anything that can be done about it (e.g. file a civil rights complaint)?
No there is none
Religious entities have a lot of rights that other organizations don’t and yes, sometimes that includes the right to do things that would constitute illegal discrimination if secular organizations did them. I’m not a lawyer and I can’t answer your question in detail, but I would suggest that you look carefully at the laws that you think protect you from discrimination on the basis of religious conviction and see whether there are exceptions in there.
Yes, religious universities have freedom of conscience. They can (within limits) impose religious requirements on students, faculty, and staff.
This is how it has to be. The alternative is to force institutions to admit people who would undermine the basic principles and existence of the school. If you are a true believer in freedom, you should support this right.
Religiously affiliated universities cannot descriminate in employment if they accept federal financial aid. This is Title VII of the Civil Rights Act of 1964. The EEOC investigate complaints.