Why Feuer sometimes and other times Füer?
Because not everybody can type ä ö ü, Duolingo lets users type ae oe ue instead, e.g. das suesse, schoene Maedchen instead of das süße, schöne Mädchen.
Unfortunately, the way it was programmed makes Duolingo think that ü and ue are completely interchangable everywhere, and that Feür is just another way of spelling Feuer.
Which is, unfortunately, nonsense. (The u and e don't even belong to the same syllable; it's Feu-er and not Fe-uer -- the eu is a diphthong pronounced like "oy".)