How about pan for man or paní for woman
In most cases you would not typically translate man as pan and woman as pani. Definitely not in this sentence. Lady and gentleman are closer expressions used for that.
I am not sure. "Žena" (woman) can be "paní" and "slečna" too. The sex are a man (muž) and a woman (žena), not Mr (pán) and Mrs (paní).
Would "Jsem já muž nebo žena?" also be accepted?
EDIT: Just went back through it again and it does appear to be an accepted answer - looking for how common this would be versus the translation above.
Not very common. Just when you really want to emphasize "já" for some reason.