"I do not have enough wine."

Translation: Nemám dost vína.

[Mad664793]

"dost vína nemám" was not accepted - is it wrong, or strange ?

[igamenir]

It is strange and not used. Maybe only in the conversation "Máš dost vína?" - "Dost vína nemám".

[cronanas]

So I'm guessing dost is like mnoho in that it requires genitive case to make sense, right?

[VladaFu]

All adverbs of quantity are like that. Dost can also be used differently (dost dobrý - quite good / good enough).