Is the Hadith: "Whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, will have the reward of a complete, complete, complete Hajj and 'Umrah" authentic? And what are the Prophetic invocations (adhkar) for this time? And how are they performed (in private or aloud)? And is it necessary to delay the Duha prayer by 20 minutes after sunrise?

The Hadith is sound. As for the remembrances (adhkar) after Fajr, Ibn al-Qayyim mentioned that the Prophet, peace and blessings be upon him, used to sit in his prayer place after Fajr until the sun rose, engaging in the morning remembrances and other forms of remembrance. He used to remember Allah at all times. Reciting dhikr silently is preferable, except for what is explicitly mentioned in the texts to be recited aloud, such as the Talbiyah during Hajj and Umrah, or for a specific benefit. The time during which prayer is forbidden after sunrise is until the sun has risen to the height of a spear, which is estimated to be about ten to twenty minutes for precaution.