Can this mean both, "I do not drink wine" and "I am not drinking wine (right now)"?
Yes
Jeg drikker ikke... vin. -Danish Dracula
So, to negate a verb, you have to add "ikke" after it.
Indeed!
How do you know when someone is saying "I don't x" versus "I am not x-ing"? Just from the context of the conversation?
Yes, it's just from context.