Does an undergraduate college student in NYC need to pay city tax?

Confused parent here. Student is moving to a shared apartment for college in New York City after been residing off-campus with parents in NY state outside the city.

For the purpose of filling out a W-4 or IT-2104 for a student part-time job with the school, how to answer "Are you resident of New York City" ?

I suppose this will affect NYC city tax withholding. Would it be different if having moved into a dorm instead of signing an off-campus lease.

At what income does the student need to file a tax return next April?
How does this affect claiming the student as dependent on the parents return ?