Not sure if this might be a reference or a coincidence, though of course "Kyougoku" is different for Maria (京極) and the fictional deity of the island (鏡獄; literally "mirror prison"):
https://en.wikipedia.org/wiki/Kyōgoku_Maria
No, Okaya does not call Maria "teacher" since as you noted, 先生 is also a title used for doctors and other professions that command respect. If he's asking whether she's a teacher, he would have used 教師 instead. In this situation, it's more like he's asking, "You're you, right?" and her response is "What are you saying? I am me..." which makes more sense leading up to Okaya talking about Kyougoku showing a person's true form.
He said そりゃ戦(や)りますよね, not やりましょう. It actually means "It's war, then." If we want to expand on this, it would be そりゃ戦をやりますよね which means "You're waging a war, then."