"I listen to music on the weekends."
So, the answer given does not have the same nuance as the English requested. 週末は音楽を聞きます makes the weekend the subject. So the English would be more like 'On weekends I listen to music.' And this would often be used to answer the question about what's done on the weekend. The English sentence is more vague, but generally we would reverse the order.
A. What do you do on the weekends? 週末に何しますか? B. On the weekends I listen to music. 週末は音楽をききます。
A. When do you listen to music? 音楽いつ聞きますか? B. I listen to music on the weekend. (Music, I listen to on the weekend.) 音楽は週末に聞きます。
So the problem is that the English sentence does not necessarily lead logically to the answer given. It would be better to change English word order to begin with "On the weekend..." to give this nuance.
This is something I've been trying to understand myself, but I think it's better to use は. Using 週末は (shuumatsu wa) says that on weekends (compared with other days) you listen to music.
It seems like you can use に, but it takes the focus away from the weekend and is just making a statement about how you listen to music and it happens to be on weekends. That's how it feels to me. It can also be used as you said to say that you will listen to music this weekend.
In my experience, 聞きます is most commonly used for both meanings (to listen, to ask), whereas 聴きます is used less commonly and usually only to mean to listen (usually to music). I'm not sure if there's a hard and fast rule about them, but that's how I've seen them used when I've read things in Japanese.
it depends on what kanji the word uses, if all kanji are part of the common kanji list it would probably be better to write it all in kanji. I think in this case duolingo didn't have the kanji for 末 in the lessons, maybe they were added later on in the second stage of the course? Idk. You should be able to write 週末 from a japnese keyboard though, on mobile or pc.