May a 13 year old boy who was just bar mitzvah make havdala for women? May he read the megilah for others?
A 13 year old that was just bar mitzvah and still hasn’t shown signs of physical maturity is considered a gadol for anything that is d’rabonon (rabbinic in nature) such as reading the megillah, but he cannot be motzei people for obligations that are min hatorah (biblical).
Regarding havdalah, it is controversial if women have to hear havdalah min hatorah or not. The 13 year old may be motzei the woman, however it is better that the woman say “boruch hamavdil bein kodesh l’chol” beforehand. This way even if she has an obligation to hear havdalah, after saying boruch… it is only d’rabonon and surely permitted.
O:CH 53-9, M:B 53-40, O:CH 296-8, Migdanos Eliyahu 2-18