Is the hadith that mentions that Jibreel informed the Prophet that angels record the deeds of the Ummah but do not record their tears true, and that a tear of a repentant person from his Ummah extinguishes the fire of Hell?

The hadiths stating that weeping out of fear of Allah extinguishes oceans of hellfire are weak, due to the unreliability of their narrators (rijal al-sanad), the presence of fabricators among them, their lack of authentication, or the possibility that they only met the Prophet without directly hearing from him.

As for the narrations (athar) from the Tabi'un and righteous predecessors that convey the same meaning, they do not constitute proof on their own.

What is authentically established concerning the virtue of weeping out of fear of Allah is what was reported from the Prophet, peace and blessings be upon him: "A man who weeps out of fear of Allah will not enter the Fire until milk returns to the udder."