What kind of music do you listen to? What kind of music do you like? What kind of music are you into? What style of music do you like? What type of music do you like? What kinds of music is fine but more common for me to use kind regardless of whether your reply 1 or more music types.

I think "kinds" or "kind" could both be correct - technically, "kinds" seems like it would be asking you to list more than one kind (e.g., I like electronic, rock, and country). Spoken naturally, it would probably come out sounding like "What kinda music do you listen to?"

2. Correct