1. This seems to be the case, else it would not need to be mentioned.
2. I think so, yes.
3. From what I'm reading, either cloister/shrine can be challenged. A challenged monk/heretic could go on pilgrimage if desired, but this would make it more difficult to win the challenge unless the opponent is also pilgrimaging (if that's even a word).
4. With regards to the Count, followers at the cathedral quarter can only be deployed to the cloister/shrine occupied by the monk/heretic. I'm inclined the same way with regards to the Flier, but not sure.
5. Since each cloister is scored individually, I think an overlapping vineyard could be scored twice.