Is it permissible to vote for a competent Christian man, or for a Muslim woman to assume leadership, or which of them is better when there is a conflict?

If all candidates are non-Muslims, a Muslim may vote for the non-Muslim who is less evil and more beneficial. However, if there is a Muslim candidate, it is not permissible to vote for a non-Muslim. The basic principle is that it is not permissible for a woman to be nominated for leadership positions or to vote for her. But if the choice is between a Muslim woman and a non-Muslim man, the Muslim woman is to be given precedence, out of regard for the name and sanctity of Islam, and because she represents a lesser harm.