Disclaimer: I’m not a teacher, but I will attempt to answer your question to the best of my ability.
The answer basically boils down to どんな = “what kind/type of” and 何の = “what (insert noun)”. That’s it. So the translation to your examples would be:
１。どんな音楽を聴きますか = What kind/type of music do you listen to? (Answer: Pop, Rock, Blues, etc) ２。何の音楽を聴きますか = What music do you listen to? (Answer: Christina Aguilera, Bruno Mars, etc) ３。どんな会社を働いていますか = What kind/type of company do you work for? (Answer: Law Firm, Music School, etc) ４。何の会社を働いていますか = What company do you work for? (Answer: Toyota, Apple, Verizon, etc)
They may seem similar, but the answers to these questions are actually different as you can see. Also, you don’t need the particle の after the word どんな. You can just put a noun right after it. I corrected that in the answers I gave you above.