In most situations, I would say to use the construction, "What's the number for X?" What's the number for the library? What's the number for Domino's? I don't see any reason to say 'phone number' because context will be clear enough. If I were to ask that question, that's how I would ask it. If you want to be more formal or polite, you can say, "Could you tell me the number for the library?"

Your question should be 'How do you ask FOR a telephone number in English?' You ask a question, you 'ask for' something that you need. And I agree that the most normal expression is the number 'for' a place or person. Do you know the number for the library? Can you give me the number for the library? and so on.

Actually I would be inclined to go with the second option, the first makes sense but is unnatural and a bit weird if used in a conversation.

I think going with "Could you tell me the phone number of the library?" would be better. You can also say, "What's your phone number?" while asking somebody's number.

what's your/his/her/someone's phone number