Short answer
Yes. Women can become scholars of Quran, hadith and fiqh, teach, and give fatwa. Aishah was among the leading jurists of the Companions, and women transmitted and taught hadith for centuries. The Hanafi school does not require a mufti to be male. What women do not do is lead men in prayer or give the Friday khutbah.
Learning what one needs for one's own faith and worship is fard on every Muslim, man or woman; deeper scholarship is a communal duty open to women. Being male is not a condition for giving fatwa. When a woman teaches men she keeps hijab and avoids being alone with them.
The four schools agree on this.
Explanation
After the Prophet ﷺ died, senior Companions brought their hardest questions to Aishah, and a large share of what we know about his life at home comes through her. Umm Salamah was consulted in the same way. The women of the Ansar were praised because shyness did not stop them from learning their religion, and the Quran told the Prophet's wives to remember and pass on the verses and wisdom recited in their houses.
The tradition continued. Women narrated hadith and taught the great scholars of later centuries, and some of the most important manuscripts of Sahih al-Bukhari were transmitted by women such as Karimah al-Marwaziyyah. In the Hanafi school, Fatimah al-Samarqandiyyah, daughter of the author of Tuhfat al-Fuqaha and wife of al-Kasani, the author of Bada'i al-Sana'i, is remembered for issuing fatwas alongside them.
Can a woman teach men? Yes. Aishah taught men from behind a screen. Today that means teaching with full hijab, in a public or structured setting, without khalwah, and keeping her voice and manner dignified. See is a woman's voice awrah.
What stays with men is leading mixed congregational prayer and giving the khutbah, for reasons explained in can women be imams. Teaching, writing, researching and answering questions are all open to women, and Muslim communities need women scholars, especially for questions women find hard to ask a man. See Aishah for her life.
The evidence
Quran, Surah Al-Ahzaab 33:34
وَٱذْكُرْنَ مَا يُتْلَىٰ فِى بُيُوتِكُنَّ مِنْ ءَايَٰتِ ٱللَّهِ وَٱلْحِكْمَةِ ۚ إِنَّ ٱللَّهَ كَانَ لَطِيفًا خَبِيرًا ﴿٣٤﴾
And remember what is recited in your houses of the verses of Allah and wisdom. Indeed, Allah is ever Subtle and Acquainted [with all things].
The Prophet's wives were charged with learning and passing on revelation.
Quran, Surah At-Tawba 9:71
وَٱلْمُؤْمِنُونَ وَٱلْمُؤْمِنَٰتُ بَعْضُهُمْ أَوْلِيَآءُ بَعْضٍۢ ۚ يَأْمُرُونَ بِٱلْمَعْرُوفِ وَيَنْهَوْنَ عَنِ ٱلْمُنكَرِ وَيُقِيمُونَ ٱلصَّلَوٰةَ وَيُؤْتُونَ ٱلزَّكَوٰةَ وَيُطِيعُونَ ٱللَّهَ وَرَسُولَهُۥٓ ۚ أُو۟لَٰٓئِكَ سَيَرْحَمُهُمُ ٱللَّهُ ۗ إِنَّ ٱللَّهَ عَزِيزٌ حَكِيمٌۭ ﴿٧١﴾
The believing men and believing women are allies of one another. They enjoin what is right and forbid what is wrong and establish prayer and give zakah and obey Allah and His Messenger. Those - Allah will have mercy upon them. Indeed, Allah is Exalted in Might and Wise.
Women share in enjoining good and forbidding wrong, which includes teaching.
Sahih al-Bukhari 3461
Sahih... بلغوا عني ولو آية ...
"Convey from me, even if it is a single verse."
The duty to pass on knowledge is addressed to every Muslim, men and women.
Where the ruling is recorded
- Radd al-Muhtar (Ibn Abidin), Kitab al-Qada, on the conditions of the mufti. Maleness is not a condition for giving fatwa.
- Radd al-Muhtar (Ibn Abidin), Introduction, on the knowledge that is fard on every Muslim. Knowledge of one's own obligations is fard on men and women alike.
Checked by our team. For a more accurate answer to your own situation, consult a scholar. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.