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...
Given that we only see what Pearce wanted us to see, it's possible he had contingency plans (even if that plan was just "Crap, need to find another exit.")
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.
During the final confrontation at the safe, when Frank/Alexander is walking to the safe to open it after Shaw threatened Elise with a knife, what does Elise mouth/whisper to him? Is it "I love you" or something else? Or is she trying to let him know the code of the safe because at that point she doesn't believe yet he really is Alexander?
Elise doesn't know the code. There is no need to hide the code from Shaw since the snipers are going to kill him anyway. She really does mouth "I love you" to him, which is why he replies "so you love me" after all is over.
The TV series
How is Elliot able to travel internationally, given his criminality and the fact that his passport has clearly been fraudulently issued?
We don't know that his passport was fraudulently issued until the second series when it's revealed that Elliot Stanley is not his real name, and in any case, a passport being used by someone working high up in an international drug-smuggling ring is likely to pass inspection at airports and the like. As for his criminal past, the Australian authorities have nothing on him as DI Rogers is revealed to have committed the murder he tries to frame Elliot for, in addition to which, Lena refused to give evidence against him; in such circumstances, they would not be inclined to subjecting his passport to closer inspection, as (in addition to there being no criminal evidence against him) they have no reason to believe that he's not who he says he is.
Is Niamh a Cassidy by blood or by marriage? In a flashback scene she is addressed as Miss Cassidy, but she has evidently retained that surname as a mature woman with two grown-up sons (one of whom was killed before the events of the show) who took that surname.
She could have retained her maiden name (unlikely in rural Ireland circa. 1980), reverted to it after he (unseen) husband's death, or (more likely, given that her sons went by the surname Cassidy) married a man with the same surname.
Going by The Reveal at the end, she's not actually a Cassidy by blood, although she was clearly raised as such.
Prior to his imprisonment of Helen, had Ruairi actually broken any laws by not reporting his wife's death and keeping her body (assuming that he is not lying about her having died of natural causes)?
