Questions that start with auxiliary / modal verbs e.g. do, have, should, are are closed questions and the answer can start with "Yes, ..." or "No, ..." Place either "," or "." after "yes" or "no". If the word "not" is in an answer to a closed question, it is because the word "not" modifies another word or phrase in the answer. e.g. Not really. Not at all. Not in a month of Sundays. Not on your nelly. Not likely. When "not" is in the answer, there may be a missing subject and verb e.g. "That's" or "It's".

“Actually, no” or “No, actually” or “Actually not”? or “Not actually”? are all heard, but "no" is more appropriate because it is a yes or no question.

What about “‘No, actually not”?

First or second options are both appropriate.