It is, although I'm not sure if "happy music" is a good translation.

I agree that happy music should be accepted as an answer too. Sounds natural to me.

yes, that's a synonym to merry, cheerful

Yes it does. (It would be Wesołej Wigilii if anybody ever said it, it is Wesołych Świąt)

But wesoły has more meanings. and the one describing holidays is used only in wishes.

Dictionary says:
1a) wesoły człowiek - feeling and showing happiness (happy man)
1b) wesoły uśmiech - showing that someone is happy (happy smile)
2) wesoły film - causing happiness (happy movie)
3) Wesołych Świąt- spent joyfully and pleasantly, only in wishes (Happy Holidays, Merry Christmas)
3a Wesołego Alleluja = Happy Easter

"they are listening to" is accepted, it works very well. Actually it's even better, I will make it the default.

Not the same thing.

It is the object, but that's not the reason why it's genitive. In fact, most verbs require the accusative with direct objects. Słuchać is one of those verbs that always take the genitive case. Here's a list of some of the most common genitive-verbs: https://forum.duolingo.com/comment/16569658 (scroll down to part 8)