La musique dont j’écoute est belle doesn’t work. It must be la musique que j’écoute…

You are confused by the nature of de in the first sentence. De is not a preposition here but a partitive article (I’m listening to [some] music) so cannot lead to dont.

Here are sentences with the preposition de and their dont counterpart:

Je te parle de la musique du XIX^e siècle.

La musique dont je te parle.

J’écoute la musique de ce chanteur.

Le chanteur dont j’écoute la musique.

But:

J’écoute [de] la musique

La musique que j’écoute.

Sometimes dont can actually be translated into English as which (or that), which otherwise is the normal way of rendering que into English.

Voilà le chat dont [not que] je suis jaloux. Why dont here? Because the valency of the adjective jaloux is de. So, you have être jaloux de. Hence the sentence Voilà le chat dont je suis jaloux = Here is the cat which / that I’m jealous of.