The word עוף has two meanings that are related but not the same. One meaning is chicken in the sense of food, and the other is a zoological classification meaning everything with feathers.

Yes. It does make sense though, since the "cook" does not only cook, they do a variety of actions. English chose cooking as the one to describe to profession. Hebrew chose butchering.

Though in modern Hebrew, a butcher is a קצב

me too

It technically is, but it's not how you would normally say it in English. Just cook, without mentioning gender.