Too late but, for me, the first is the correct one. The second one sounds like a big feast every year or maybe a competition where people eat a moose, for example, "eating-the-moose tournament" or "eating-the-moose time!". So, the correct way for me to say it'd be "What is the moose <--(at the beginning you're placing all of what you're going to ask about (subject) ) eating?".
The sentence translates to What does the moose eat?. I believe the reason your answer was marked as incorrect was because you missed the article 'the' before moose.
[en elg = a moose, elgen= the moose]
About the second question, I am not very sure about the answer and would ask you to check. Google translates the plural to " hva spiser ikke elg". I don't know whether it is correct or not, but the Norwegian course did tell us that ikke meant 'no [negative]'. I would like an Advanced Norwegian learner/ speaker to verify this. :)