Ilya Kabakov's painting is titled 'Whose Fly is This?'. It reads: "Anna Evgenievna Koroleva: Whose fly is this?; Sergei Mikhailovich Khmelnitsky: That's Nikolai's fly". The object in this case is a painted representation of a fly. Just as a fly can belong to no one, or is so insignificant that no one would care to claim it (except apparently Nikolai), the painting belongs to no one. Its original owner has discarded it and only an anonymous Garbage Man, or a Man Who Never Threw Anything Away would think to try to identify its owner. (http://www.3dlit.org/practice/kienholz_kabakov/Kienholz_Kabakov.html)
Stupid...sure, but it serves a pedagogic purpose. It first makes a grammatical point. Then, by being factually stupid, funny, weird, implausible, etc., it gets you to think about it, and that makes you more likely to remember it. Hey, DuoLingo, banish the banal, strike the stale and trite, and bring on the weird and wonderful examples. I love them.
Another reason could be that it is an example of how to use the principle of the different spellings in sentences when you ask a question. Also, it serves as incentive for me personally. I always come into these with pen and paper waiting for the next hilarious statement to be made.
You could say it like that. It would sound weird, but still correct.
Also: "Это чья муха?" (sounds normal), "Это муха чья?" (reaaaally weird), "Муха это чья?" (less weird), "Муха чья это?" (normal; gives off a feeling like it isn't the first time someone asks whose fly it is but they still didn't get an answer). Not sure if Duolingo will still accept all those variants, but they all are correct in Russian (even if some sound really weird).