Did the Prophet (peace be upon him and his family) inherit Lady Khadijah (peace be upon her)? In what did he spend her wealth? Did he give any of his wealth as a gift to anyone, and what was the reason? And who are "those whose hearts are to be reconciled"?

It is not certain that Lady Khadijah, may Allah be pleased with her, left behind inheritable wealth after her death. It is probable that she spent all her wealth in the cause of Islam, just as Abu Bakr Al-Siddiq, may Allah be pleased with him, did. Some scholars have mentioned that Khadijah, may Allah be pleased with her, had no wealth because she had gifted her wealth to the Prophet, peace be upon him, during her lifetime. It has been reported that the Prophet, peace be upon him, used to accept gifts and reward for them. Examples of this include his story with Abdullah ibn Umar when he bought a camel from him and then gifted it back to him, and similarly his story with Jabir ibn Abdullah, may Allah be pleased with him, when he bought a camel from him, gave him its price, and then returned it to him.