I know this one from "Skam"... ^^
Is the h in hjemme silent?
The 'h' is always silent before another consonant.
Is it like that in all dialects? I kinda got used to saying hjemme with that /ç/ sounds at the start.
Is this the instrumental case?
Is hjemme a noun?
No, it's an adverb
is hjemme the adverb form and then hjem the noun?
Yes and no... "hjem" can be an adverb, you use it when indicating a direction. "Jeg drar hjem" = "I am going home" vs "Jeg er hjemme" = "I am at home". But there is also the noun "et hjem" = "a home".
Would it be correct to say "Is she home", or does it have to be "Is she AT home"?
The two English sentences seem pretty interchangeable to me, so they should both be correct