"Must I pay?" has a different feel than "I must pay?". The former is resistant and defensive. The latter is surprised and whiny.
It may also mean a genuine inquiry. Or would that be "Sollen ich bezahlen?" Should I pay?
(In the example you offered, it should read "Soll ich bezahlen?" not "Sollen ich bezahlen?" because you have to conjugate the modal verb ("sollen") while leaving the other verb ("bezahlen" unconjugated. :) ) As for your question, I do not know the answer. Sorry...
Well, that would be closer to "Ich muss bezahlen?" than the given German sentence. It's better to stick as closely to the given German as possible, because our little green owl is picky ;-)
Same here. In English you can leave off Do at times as long as you have a question mark. If we always require "Do" than we can just abolish the question mark.
must should also be incorrect based on meaning because it emphasises an inner motivation which is obviously not the case in here. At least that is what I remember. Even if it were correct, have to would be better imo. correct me if I am wrong though.
In the US, I hear must and have to used interchangeably, in informal speech. But, in informal speech, have to is more common. In formal speech, must would be more common.
So in other words, you can use either one, but if you're watching American movies and TV you will probably hear have to much more often.