"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 it is used!
Because Duolingo says the stressed word is normally has the last position in Czech sentence. So using this rule "nemám" in the end - is just perfect position...
It's rarely used. It's almost like saying "Enough wine is something I don't have". It could be accepted but then it would likely lead learners to believe that you can use this word order any time.
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).