When is the Sunnah of Fajr prayed, is it after the first Adhan or the second Adhan? And is praying between the two Adhans considered part of Qiyam al-Layl? And is "Quran al-Fajr" recited after the Fajr prayer or before it?

The time for the Fajr Sunnah prayer begins with the appearance of the true dawn (usually at the time of the second adhan), and it is not valid before it. The period between the two adhans is among the best times for night vigil (qiyam al-layl) and for prayers to be answered. A Muslim should delay the Fajr Sunnah prayer until after the final adhan, though he may perform voluntary prayers before that. Congregational prayer is obligatory, and he must go to the mosque to attend the Fajr prayer in congregation.