I would have translated this into: "are you taking..." because, in my understanding, the Swedish sentence doesn't imply that the asker will be at the party.
Yes - is there no difference between 'taking to' and 'bringing to' in Swedish?
Sometimes they can be interchangable but bringing feels like something is getting closer to the speaker and taking feels like it is getting further from something. It's a bit like the difference between here and there but there are lots of examples where it'd be unclear which to use. When you physically bring something closer to your body in order to transport it away from something, you are sort of bringing and taking simultaneously. So, it's really dependent on the perspective of the speaker and how they imagine the event.
That's what I mean - in English we differentiate by having two different words, and I'm wondering whether Swedish also has two ways to express it, or just one.
That is correct. Both are correct.
Usually you'd say that with emphasise on the wine.
Imagine someone looking into your bag, and with surprise he asks "tar du VIN med dig till festen?"
"Tar du med dig vin?" is asking if you're bringing wine in general.
I assume you can because you can say "Har du en klänning på did?" and "Har du på dig en klänning?"
The sentence contains neither ’your’ nor ’some’, it’s just plain ’wine’.
I wrote "Do you bring wine for the party?" and duolingo said it's wrong.
Can't 'till festen ' be translated as 'for the party'? If then, what is the correct/natural swedish expression of it?
My first thought for the sentence is two people discussing a party later that night to which both will be going. In that case it would be "to", as duo suggests.
However, it could be for example that A (the speaker) want B to bring some of his wine to A, that he is about to visit, so that A can serve that wine at a party A is throwing later, and to which B may or may not come. In that case it would be "for" as you say, but then I think the sentence must be about the future and would be "will you bring [me] wine for the party?". That can be said as the Swedish phrase is now (or as "kan du ta med dig vin till festen?" for "could you bring...").