Why does Hayate have to cover herself when she enters and leaves Saint's Church?
It is probably customary, just like it is a custom in some Christian churches that a woman is expected to cover her head when entering a church building, especially during the services.

The Sankt Kaiser Church is modeled after Christianity in general and the Catholic Church in particular. It also has an autonomous area on Mid-Childa; compare the Vatican city state. (There even exists a counterpart of the Turin shroud.)