So are feeding the hungry, and housing the poor, but if there were church-sponsored supermarkets and subdivision developers, would you expect them to be exempt from the laws that govern Safeway and Donald Trump? I'm not against letting a church impose its tenets on workers who do purely charity work as part of that faith's mission in the world, often the workers in such an institution are not "open market" employees anyway, they're people who generally make subminimum wages (or close to it) who derive a substantial part of their compensation by serving their spiritual needs. You just can't say the same thing about the average radiologist who works for a Catholic hospital, or a sex education psychology professor who works for Notre Dame. Often, these employees are not of the Catholic faith, and they have been recruited in the labor market, and been lured away from other institutions, private and governmental. The previous type of worker mentioned is doing the work as a large part of his or her faith, and is a direct representative of that faith. I don't expect a dietician at a Catholic hospital to refuse to provide me a meat choice on Good Friday, but I would expect a Catholic soup kitchen to provide meatless meals on that day.
I acknowledge that it might be difficult to separate out core functions from peripheral business functions, but the best way to sort them out is to ask: Is the employee delivering the services with the prime mission of advancing the belief system of the religious organization that heads the institution? A priest, a nun, a brother in a religious order, or a deacon probably certainly would be; whereas a janitor, an instructor, or a nurse would probably not be, if they were not members of religious orders.
Irrelevant. The mission of the organization does not depend on the motivation of the employees. As long as the work gets done the mission advances.