Someone consult St Thomas Aquinas.
RE: Schizophrenia... Depends whether he has psychosis or not. You can get psychosis, and then go back to normal in time, but the diagnosis remains. It ought not to disqualify anyone from any position. The pope having extreme psychological tendencies would not only be imitating the greatest saints, but would be willing to step outside the constraints of the real world to practice his faith, that would be an example to us all that this world is not as important as the faith. Also, if he is willing to step beyond reason motivated by his faith he will make a most excellent pope. The church fell because everyone who could have made a difference was concerned about their own welfare, their position, etc. If the pope didn't care what would happen to him he would imitate the martyrs and would be a true saint, and the church would have a fearless leader. Btw, it is proven that there is no such thing as schizophrenia, there is no scientific test to prove it exists, that disease is the sacred cow of psychiatry, even its discoverer admitted that it didn't exist and could be a psychological extreme state. Anyone who has the faith deeply enough will step beyond cultural morality and will appear to abandon reason, but their interests will be the sanest thing of all, winning the approval of God.
The "mentally ill" are often very religious. I am one, as everyone knows.