Can you also say 'Jutro będzie padał śnieg.'?
Yes. Makes no difference in this case.
"Tomorrow snow will fall"?
I can never hear the first p in a word. Is that the fault of the app, or me, or just the way Polish works?
A mix of the first two, perhaps? Sounds fine to me here. Not the way Polish works, definitely. No silent letters.
Tomorrow there will be snow?
Hmmm... close enough, added.
Why was it not accepted "There will be snowing tomorrow"?
I don't believe it's grammatical in English.