Learn English – use “is” twice in the question starting with “Is”

Maybe you could say it like this:

The teenager couldn't help asking him.

"Where did you meet this Frank?" Inquired his young customer curiously.

Repeating words it's not good and you should avoid it as much as possible. But it depends, if you're writing something informal don't be too concerned. If it's formal, or a story, you should try to put different words. A dictionary of synonyms could help a lot.

Are you required to hold a work permit in this country?

In this sentence, are should be used because it agrees with the second person pronoun "You". You only need a do in a question when the sentence lacks of auxliary verb, as in

Statement : You hate me (no auxiliary)

Question : Do you hate me? (do is added)

Are there any water or is there any water?

"Are there any water" is wrong, water is uncountable, so it takes a singular verb.