"What is under the hospital, a store or a bank?"
Translation:Mi van a kórház alatt, egy üzlet vagy egy bank?
mit has the accusative ending -t, so you would use it when asking about the object of a sentence.
For example, Mit látsz? "What do you see?*
The "what" is the object of the seeing, the thing that is seen.
In this sentence, though, you are asking about the subject -- the thing that is doing the "being somewhere".
What's the difference between lent and alatt?
alatt is a postposition. You use it together with a noun.
lent is an adverb. You use it on its own.
It's much like the difference between "below" and "at the bottom" -- you can't say that something is "at the bottom the hospital" (without "of"), nor can you use lent with a noun.