"¿No cocinas pato?"
Translation:You do not cook duck?
It is correct, if you are speaking generally (not about a specific bird). "I don't cook duck" and "I don't cook ducks" would both work and be understood. There is a slight difference, though: "duck" would be talking about the food and "ducks" about the animal. "I don't cook a duck" sounds awkward because the article is usually dropped in English.
“I don't cook duck/ducks" is not correct English in most cases. That would only be correct if you were quoting/mocking/characterizing somebody else's sentiment. Do I not cook duck or do I cook duck would both be correct, depending on what you are trying to say. But duck here refers to a type of dish (like saying “Do you eat bread, regardless of how many loaves or how many styles of bread). It doesn't refer to the number of individual animals being cooked.
I'm sorry but you have this completely wrong.
"I don't cook duck" is perfectly correct in English and is not in any way mocking anyone. It is simply a way of saying that you do not cook duck (the food).
"I don't cook ducks" would be a strange thing to say as it would be referring to live ducks and wouldn't refer to food in most cases.
"Do I not cook duck" is completely wrong and does not make sense. "Do I cook duck" is very strange, as who would ask themselves if they cook duck?
This is correct, though, "Do I cook duck?" could be used in a conversation to make sure that you understood what someone was asking you (the original question possibly being, "Do you cook duck?") Something like, "Do I not cook duck?" would only really be seen in dramatic situations like plays/movies/shows where a character could, for example, be a renowned chef known for his duck specials. It would be a rhetorical question, meant to be a sarcastic "yes" to someone questioning his skills or abilities.
These, however, are special cases in English that are not correct in typical, informal conversations generally.
They are all strange things to say.
I should have included the "?" but that should have been obvious based on what I replied to. "I don't cook duck?" is not "perfectly correct" in English unless you are quoting or mocking.
"Do I not cook duck?" is strange, but it is what "Don't I cook duck?" means. Strange? Absolutely. That doesn't make it incorrect.
I don't know why you assumed any of these questions were being directed at oneself. That is not implied in any way.
I thought "Do I cook duck?" was directed at oneself as I couldn't think of a situation where you would ask someone else that question - how are they meant to know if you cook duck? It's an answer that really only you, yourself would know. But thinking on it, it could be said to clarify what someone has asked you as Mark mentions above.
I get what you're saying with the "I don't cook duck?" now. I didn't notice the question mark in the post you were replying to so I agree on that one.
Changing a word to match gender only happens with nouns and adjectives. With verbs, you change the ending to match only the person who is performing the action. (gender excluded) For example: Yo hablo and ella/él/usted habla. It matches the speaker not the gender.
This was probably confusing but hope it helped a little!
It depends on the context. You're absolutely right that "estás cocinando" is the present progressive form, but Spanish often uses the present progressive when English would use the simple present:
You don't cook duck [in general]? -> ¿No cocinas pato? You aren't cooking duck [right now]? -> ¿No estás cocinando pato? You aren't cooking duck [near future]? -> ¿No cocinas pato?