Are there established virtues from the Prophet, may God’s prayers and peace be upon him, concerning the remembrance of God (dhikr) and deeds performed between the Sunnah prayer of Fajr (Raghibat al-Fajr) and the establishment of the Fajr obligatory prayer?

No actions have been reported from the Prophet, peace and blessings be upon him, after the two rak'ahs of Fajr prayer, except lying down on his right side. This is the view of the Shafi'i school, contrary to most jurists.

As for supplications (adhkar), Ibn Khuzaymah mentioned in his Sahih a lengthy supplication from Ibn Abbas that the Prophet, peace and blessings be upon him, used to recite after the two rak'ahs of Fajr, but its chain of transmission is weak.

Nevertheless, it is confirmed that supplication is not rejected between the adhan and the iqamah, so one should increase in it during this time.