"I can have a cookie, can't I?"
(Please ignore the double quotes while reading)
What is this type of question called?
Also, is it grammatically correct under American English?
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That's called a tag question.
As far as I know, it's grammatically correct in many variants of English, American English included.
A question tag or tag question (also known as tail question) is a grammatical structure in which a declarative statement or an imperative is turned into a question by adding an interrogative fragment (the "tag"). For example, in the sentence "You're John, aren't you?", the statement "You're John" is turned into a question by the tag "aren't you".