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Just seconds before Frank approaches the safe, he says, "I see" to which Elise replies, "Now you see." What do they mean and what do they see?
Just how out of his league Frank actually is, would be my guess.
A guy in the hotel seems to be unhappy with Elise checking in with Frank. Why?
Because she's hot and he was hoping she would be alone.
Why did the chief operations officer fire Elise, and more importantly, why did she thank him?
Not sure why he fired her, but Elise wanted out of the police so she could be with Alexander. Why she would say that out loud I don't know, probably the officer just thought she was being polite.
She was probably fired for incompetence, since this mission to catch Pearce had been an almost complete failure and because of the suspicions about where her loyalties truly lie. She thanked him because, when you send an international sting operation completly tits-up, you are usually lucky to end up merely unemployed rather than facing charges yourself.
What did Frank get out of the safe? It looked like a checkbook, which wouldn't be so precious or worth fighting for.
A checkbook, yes, plus other documents related to the money. Those would be worth fighting to recover. Plus, the whole thing was mainly a gambit to get rid of Shaw.
Did Elise want Frank to get into her room in the hotel when she realized he was standing in front of the door?
Frank/Pearce's entire plan relied on Elise choosing "a man of [his] height and build" from all the men on the train. Suppose she had chosen someone else? Then what?
Or if Interpol had arrested Frank upon arrival in Venice? He was saved only because Acheson managed to read the burned note completely...
Frank/Pearce getting captured by the police/Shaw. What was his plan to get out of that mess? Was he counting on Elise liking Frank enough to rescue him?
Pearce's plan seems to have been an extremely sketchy Batman Gambit, with the gaps filled in by a series of Indy Ploys. If he had the balls to rip off a man like Shaw and think he could get away with it, he had the balls to try and bluff his way through just about anything else that came up.
Elise seems to be quite rich, while her occupation is undercover agent. Is this job really so well paid?
She went undercover as the lover/wife of a billionaire. That'll tend to expand one's available resources.