"Do you leave? When?"
Translation:An imíonn tú? Cathain?
Imigh is the action of leaving a location, whereas fág is leaving an object someplace. Like, "I leave the restaurant." would be "Imím an bhialann" because it's a place, not "Fágaim an bhialann" because you can't exactly place the restaurant somewhere. In the same way, "I leave the pen on the table" is "Fágaim an peann ar an mbord" because you can move the pen, but not "Imím an peann ar an mbord" because the pen isn't a location.
Without understanding the full context of your confusion, the simple answer is that you use an bhfuil if the question is an "Am/are/is?" question - in other words the verb is "to be", or bí in Irish (ignoring the copula for the moment). The verb bí is an irregular verb. One of the ways that is irregular is that, in the present tense, you have tá to make a positive statement, an bhfuil to ask a question, and níl to make a negative statement. Níl is derived from ní fhuil, so you can see that an bhfuil and níl are related.
In Irish, where you answer a question by repeating the verb, the answer to an an bhfuil question will be tá or níl. (This is very briefly referred to in the Tips & Notes for the Present Tense skill).
In this case, the verb is "leave". (English is a bit broken in this regard, as it uses an auxiliary verb "do" to ask this type of question, so you actually have 2 verbs, and you can answer "Do you leave?" with "I do", "I leave" or "yes").
If the question was a "to be" question, like "are you leaving?", you would use an bhfuil tú ag imeacht?. But because it is a "leave" question, you use the interrogative particle an with the verb imigh - an imíonn tú?