"Un cuisinier" is a chef or a cook. I think you meant to ask about "cuisiner" - "to cook".
"Faire la cuisine" is about the action of cooking. It is never followed by what you're cooking, since it basically means "to make the food".
"cuisiner" can be transitive (followed by what's being cooked), or intransitive (just the action itself). Only the intransitive use is interchangeable with "faire la cuisine".
I wouldn't.... https://www.collinsdictionary.com/dictionary/english-french/a-female-rat (don't know how trustworthy this is though)
Don't all animals have masculine and feminine forms because there are male and female animals?