I think in English you'd say "Do you have a light?" accordingly. "Hast du Feuer?" isn't an abbreviation of "Feuerzeug", but refers to the actual fire coming out of a lighter or from a match.
I think there are words that get shortened in German, like "Auto" for "Automobil" (but nobody says "Automobil" anymore anyway, so it doesn't really count), or "Defi" for "Defibrillator" (but I assume that's just an abbreviation medical staff use to speed up the conversation), but I can't think of any [proper ones] at the moment. Maybe you'd like to ask this question in a new thread, there might be some interesting answers.
But why «mein» does not become «meine» in the second case?
Because meine comes before grammatically feminine or plural nouns.
Werkzeug is grammatically singular, even if it refers to several tools.
Compare the English word "equipment", which often refers to multiple items at once but is nevertheless always grammatically singular. You would say "My equipment is new" and never "My equipment are new".
Same idea here.
Feuerzeug is indeed the correct spelling.
Because many learners do not have ä ö ü on their keyboards, Duo lets you type ae oe ue instead.
Unfortunately, Duo thinks that the reverse is true as well, and that ue can always be replaced by ü, leading to spellings such as Fraün or Feürzeug.
Ignore those if you can.