Are the two rak'ahs of greeting the mosque sufficient as a substitute for the pre-dawn Sunnah prayer for someone who prayed them doubting the entry of the time for Fajr, then the prayer was established? And should the Sunnah prayer of Fajr be made up after the obligatory prayer or after sunrise?

Whoever prays تحية المسجد (the salutation of the mosque) and then the Fajr prayer is established before he prays the regular two rak'ahs (ratibah) of Fajr, he should pray them after the obligatory prayer. This is based on the hadith of Qais bin Fahd, who prayed them after the morning prayer, and the Prophet (peace be upon him) approved of it. It is, however, better to pray them after the sun has risen and ascended by the length of a spear, due to his (peace be upon him) saying: "Whoever did not pray the two rak'ahs of Fajr, let him pray them after the sun rises."