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
Okaya calls Maria teacher even though she is the nurse. That is because sensei has deeper meaning in japanese than just teacher. It generally means someone of higher rank who can teach you something. Doctors are also called sensei, but i felt like it is the most organic to translate it as teacher.
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.
The last panel features Okaya saying "Let's do it". While yarimashou means let's do it, this time a special way of typing was used. The character used in this word was for the word tatakau / fight. This is done to give the word double meaning. It means both let's fight and let's do it at the same time, even if the character technically has the wrong reading.
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."