Is the Fajr prayer performed alone at home after sunrise and after the Adhan and the two Rak'ahs of Sunnah valid, and if not, what should be done?

Whoever misses a prayer due to sleep or forgetfulness must pray it immediately upon waking up or remembering it, as the Prophet, peace and blessings be upon him, said: "Whoever forgets a prayer or sleeps through it, its expiation is to pray it when he remembers it," and his saying: "There is no negligence in sleep; rather, negligence is upon him who does not pray the prayer until the time for the next prayer comes." Scholars have agreed upon the obligation of making it up. What you did concerning the adhan (definition), sunnah (definition), and iqama is correct and in accordance with the sunnah of the Prophet, peace and blessings be upon him.