Well, that still makes sense though, but 요리사가 요리를 잘해요 is way much frequently used.

I believe it doesnt work. Similarly you couldnt say dont cook by putting 안 before 요리. It needs to go 요리 xx 해요 if it's modifying the verb.

공부 안 했어요. I didn't study. You couldnt put 안 공부했어요. I think it would also be 요리 못 해요. For I can't cook.

I'd also would write it like that but we should wait if someone knows it better

Also wondering...