You are correct – 커피(를) 안 좋아해요? means “You don’t like coffee?”
I would like to point out that there is a difference between being formal (or informal) and polite (or impolite/rude).
먹어 – informal and potentially rude (if used improperly)
먹어요 – informal but polite
먹습니다 – formal
좋아해요 therefore is informal but still polite – perfect for most situations in Korean, and a safe bet for a foreign learner of the language to use. On the other hand, 좋아해 is both informal and impolite and should be used with caution – to children, to animals, to yourself, or to others below you status-wise. Otherwise, it will sound rude. I mean look at who it’s meant to be spoken to – using it with the president of a company would be like treating him as if he were a child, animal, or unimportant person.
Replying with 네 would be incorrect, as it would mean you were agreeing with the speaker and that you do not like coffee after all.
There’s nothing wrong with just answering yes or no, restating the question, or restating the question with natural omissions. It might seem boring when learning a foreign language, but it’s quite natural. You don’t always need a long, elaborate answer. I mean, natives don’t answer that way. It’s just because you’re learning and you want to use what you know and not just sound like you’re copying or clueless. But truth is, sometimes that’s just the way a good response goes
You could respond “아니요”; “아니요, 커피(를) 안 좋아해요”; “아니요, 안 좋아해요”; “아니요, 안 좋아요.” I would pick the third one or just the first haha.