I disagree; another way of saying it is "When do you finish by?" but some people do not like ending a sentence with a preposition ("by") so they rearrange the sentence to "By when do you finish?"
"Before you finish?" is more like a suggestion that can be answered with an agreement or disagreement whereas "By when do you finish?" asks for a specific time.
"Quando finite?" would be enough for that; "entro" specifies that the answer must be an upper boundary, i.e. if I plan to finish between 10am and noon, I'll answer "entro mezzogiorno" (kind of "within noon") or "per mezzogiorno" (by noon). Given the renown Italian punctuality, asking vague questions like this is usually for the better :P
You know, you make a fine point. Duolingo's trying to teach you the adverb "entro", but it's true that it would make sense with the verb as well: it's very colloquial, but it can mean "should I enter when you finish?" (not may). I wouldn't say it's the correct translation, but it should be accepted.