Did Abu Talib die in disbelief or in submission (Islam), and how is that, given that his wife, Fatimah bint Asad, embraced Islam? And how did the Prophet, peace and blessings be upon him, allow her to remain married to him?

Abu Talib died as a disbeliever, by the consensus of the people of knowledge. The Prophet (peace be upon him) advised him to say, "There is no god but Allah," but he refused, fearing the scorn of Quraish. Thereupon, the verse was revealed: "Indeed, you do not guide whom you like, but Allah guides whom He wills. And He knows best those who are [rightly] guided" (28:56). The hadiths mention that Abu Talib is in a shallow pool of fire, and he is the least punished of the people of Hell. As for the Prophet's (peace be upon him) allowance for Fatimah bint Asad to remain married to Abu Talib, this was before the prohibition of believing women remaining under disbelievers. Abu Talib died three years before the Hijrah, while the prohibition was revealed during the Treaty of Hudaybiyyah in the sixth year of the Hijrah.