Burping does not break the fast, but if some food comes out with it, then it must be spat out. If it is swallowed deliberately, then this breaks the fast.

Listening to music is haram, whether in Ramadan or at other times, but it is more forbidden in Ramadan, and is a greater sin. For more, please see the detailed answer.

If you cannot help vomiting, your fast is valid. Also, swallowing vomit unintentionally does not break the fast.