Is it permissible to demolish the mosque wall and rebuild it so that the grave is outside of it, and will prayer in it be permissible thereafter, along with a refutation of the misconceptions related to praying in a mosque that contains a grave?

It is forbidden to build mosques over graves and to pray in them. This is due to the explicit prohibition of such acts in the Prophetic Sunnah, and because it resembles the actions of the Jews and Christians who took the graves of their prophets as places of worship. Furthermore, singling out graves for prayer at them resembles the veneration of idols.

The Prophet, peace and blessings be upon him, cursed those who took graves as mosques or lit lamps over them, due to the waste of money involved and the excessive veneration of graves, which could lead to polytheism (shirk).

As for the argument of some concerning the presence of the grave of the Prophet, peace and blessings be upon him, and his two companions in the Prophet's Mosque, the response is that the mosque was not built over the grave. The Prophet, peace and blessings be upon him, was buried in his house, and the inclusion of the Prophetic chamber into the mosque was not by consensus of the Companions. Moreover, the grave is not in the mosque itself but in a separate chamber, independent of it and surrounded by walls.

Any structure built over a grave must be removed, or the grave must be separated from the mosque if it is found within it. As for slaughtering, vows, supplication, and seeking help from the deceased, these are major polytheism (shirk) because they are acts of worship directed to other than Allah.