Did Abu Talib die as an unbeliever or a Muslim, and how did the Prophet, peace and blessings be upon him, allow his Muslim wife to remain married to him?

Ahl al-ilm (the people of knowledge) consider Abu Talib to have died a disbeliever. The proof for this is that the Prophet (peace be upon him) invited him to Islam at his death, but he refused. Then the verse was revealed: "Indeed, you do not guide whom you like, but Allah guides whom He wills" (Al-Qasas: 56). Authentic hadiths have indicated that he is in a shallow pool of fire, from which his brain boils.

As for Fatimah bint Asad remaining married to Abu Talib, this was before the prohibition of believing women staying married to disbelievers. This is because Abu Talib's death occurred three years before the Hijra, whereas the prohibition of believing women remaining married to disbelievers was revealed in the year of Hudaybiyyah, which was six years after the Hijra.