"dost vína nemám" was not accepted - is it wrong, or strange ?
It is strange and not used. Maybe only in the conversation "Máš dost vína?" - "Dost vína nemám".
So I'm guessing dost is like mnoho in that it requires genitive case to make sense, right?
All adverbs of quantity are like that. Dost can also be used differently (dost dobrý - quite good / good enough).