What are the evidences from the Quran and Sunnah used to respond to those who support women leading men in prayer?

It is not permissible for a woman to uncover her face and hands in front of non-mahram men. This is based on the general evidence mandating covering, and because uncovering them often leads to fitna (temptation/strife). This is the view of the majority of jurists.

Among the proofs for the obligation of covering are: Allah’s statement, {O Prophet, tell your wives and your daughters and the women of the believers to bring down over themselves [part] of their outer garments. That is more suitable that they will be known and not be abused. And ever is Allah Forgiving and Merciful} [Al-Ahzab: 59], and His statement, {And not expose their adornment except that which [necessarily] appears thereof and to draw their headcovers over their bosoms} [An-Nur: 31].

There is also the hadith of Aisha, that Asma bint Abi Bakr entered upon the Messenger of Allah (peace be upon him) wearing thin clothes, so the Messenger of Allah (peace be upon him) turned away from her and said: "O Asma, indeed when a woman reaches the age of menstruation, it is not proper that anything be seen from her except this and this" - and he pointed to his face and his hands. This hadith is weak according to the Huffaz (masters of hadith), and Al-Albani's authentication of it is unsound. Even if the hadith were authentic, it would be interpreted as applying to the state of prayer. For awrah (parts of the body that must be covered) in prayer is one thing, and awrah in front of non-mahram men is another.

The Sunnah indicates that a woman's face is not awrah in prayer, as in the two Sahih collections, from the hadith of Ibn Umar from the Prophet (peace be upon him) who said: "The woman in ihram should not wear a niqab (face veil) nor quffazayn (gloves)." This indicates that a woman's face and hands are not awrah, because the Prophet (peace be upon him) forbade the woman in ihram from covering her face and hands, and if they were awrah, it would not be permissible to uncover them. This indicates that the hands and face are not awrah in themselves, but it is legislated for a woman to cover them in front of non-mahram men.

Shaykh Al-Islam Ibn Taymiyyah said in Majmoo' al-Fatawa: "For this reason, he did not forbid uncovering them - meaning in ihram - rather he forbade her from the niqab and quffazayn; because that is clothing for the face and hand, not because it is awrah. Rather, the hand, foot, and face are not awrah in prayer, but it is not permissible for a woman to go out without a jilbab, and she should draw her khimar over her bosom, and she should not display her adornment except to her mahrams, or what appears thereof unintentionally, such as what cannot be concealed."

Uncovering the face and hands is not permissible according to the majority of scholars, because all of them are adornment, and Allah the Almighty has commanded to conceal all adornment except what appears thereof. Ibn Mas'ud and Ibn Abbas (may Allah be pleased with them both) said that the apparent adornment is clothing, and it was also said that it is what appears unintentionally.

The widespread balwa (affliction/common practice) among some people does not permit the uncovering of the face and hands.

Therefore, it is obligatory for a Muslim woman to cover her face and hands, because they are adornment, and she has been commanded to conceal all adornment, and because uncovering them is a cause of fitna.