Les rêves des amoureux sont comme le bon vin, ils donnent de la joie ou bien du chagrin...
"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".
Why was ". . .can do the cooking" not accepted? (I know nothing about the film people are referring to, so understand there may be an idiom here, but just as an accurate translation, it seems at least as good.)
"Can do the cooking" doesn't flow very well in English... While it's technically correct it would not be said in this way
N'importe qui peut faire la cuisine!
Could "le rat peut cuisiner" work as well?
Only if you change the sentence. Ce means "this" specifically. While le means "the". So to answer your question, no it would not mean the same
Well, yeah. Otherwise it would be "cette," right?
Oh, were you being sexist?
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?
Les rats sont des animaux très intelligents et très sympathiques. Je les aime. Et je les adore quand ils font la cuisine !