Is reciting Al-Fatihah after finishing a speech or supplication legislated (in Islam)? And did any of the Salaf (early Muslims) do it?

Raising the hands and reciting Surat al-Fatiha after lectures or supplications has no basis in Islamic law and is considered an innovation (bid'ah). As for the listeners weeping out of emotion from stories, or the preacher weeping, there is no harm in it. Indeed, Abu Bakr and Umar (may Allah be pleased with them both) wept when Umm Ayman wept, and the Companions and the Messenger of Allah (peace be upon him) were moved by admonitions and verses.