Is it confirmed that the Prophet (peace be upon him) prayed two rak'ahs after Fajr prayer, immediately at sunrise, even though this time is a time when prayer is forbidden? And is it obligatory to regularly perform this prayer immediately at the time of Ishraq, or is it safer to omit it?

The prohibition of supererogatory prayers (nafl) at sunrise is established by authentic hadiths, such as the hadith of Amr ibn Abasa al-Sulami. The jurists have agreed on the reprehensibility of absolute supererogatory prayers after the Fajr prayer until the sun rises. Ishraq prayer takes place after the sun has risen to the height of a spear, and it is a recommended Sunnah, not an obligatory one.