Improper translations: "Whose is the coat" or "whose is the shirt" as a translation of "À qui" (being a part of a sentence). Generally I believe you cannot end a sentence in a preposition. Therefore, the translation should read, "Whose coat is it?" or "Whose shirt is it" or rather "To whom does that shirt belong?"
Technically, you are correct, but having a preposition at the end of a sentence has been acceptable in modern English for some time now.
Yes, I think there's some confusion in your question. In "Whose is the shirt" the sentence ends in a noun. This is correct English. But I agree with Respheal that in current usage a preposition can be fine.
Even if you were to end a sentence with a preposition - this doesn't validate the duolingo expression - "whose is the shirt"
I've never heard "Whose is the shirt?" used in proper English diction. But I could certainly be wrong.
In spoken English you may hear "whom does that shirt belong to?" - then it indeed finishes with a pre(post)position.