"Do you want to listen to music now?"
Translation:¿Quieres escuchar música ahora?
If you are talking about specific (definite) music, you would use the definite article. Do you want to listen to the music (that is playing right now)? Here you are not talking about specific music but any music so the article is n ot used.
And you do not use "a" because eschuar does not mean just to listen, it also means to listen to.
Duo will often use a variant of the correct answer when you've changed a significant word in the answer. If you wait for the question to come up again and just hit skip.. or "check" without answering, it will show you what they have as the real correct answer for it. Happens to me frequently if I use need instead of want, etc..
"¿Tú quieres?" or "¿Quieres tú?" would be OK but the tú would be redundant as it is the only subject pronoun that can go with quieres. Unless you want to use it for emphasis most speakers would drop the pronoun tú. I believe Duo is looking for the most common answer and are trying to get you in the habit of not using it when it is redundant. However, as you know by now, Duo is not always consistent. But I am glad they are around.