If a woman attends Friday prayer, should she intend the فرض (obligatory) prayer of Jumu'ah or not? And is she required to re-pray Dhuhr afterward? And is it recommended for the Friday Khateeb (preacher) to address women explicitly by saying "O sisters," or should he suffice with saying "O brothers" to include everyone?

Jumah prayer is not obligatory for women by scholarly consensus. However, if a woman performs it, it suffices her for the obligatory Dhuhr prayer, also by scholarly consensus, because Jumah is more complete than Dhuhr. When a woman prays Jumah, she should intend what others intend, meaning she intends the Jumah prayer. According to the preponderant opinion, she is not required to intend it as an obligatory prayer; merely specifying the prayer is sufficient. It is not prescribed for a woman to repeat Dhuhr after Jumah. As for the preacher addressing the congregants, there is no evidence specifying a particular wording for women, and the matter in this regard is broad.