If someone enters the mosque before the Fajr prayer is established, should he pray two rak'ahs for greeting the mosque, then the Sunnah of Fajr, or is it sufficient to pray the Sunnah of Fajr, and it counts as greeting the mosque?

The two Rak'ahs of Fajr prayer are sufficient for themselves and for the greeting of the mosque (Tahiyyat al-Masjid), and this is preferable. It is also permissible to pray the greeting of the mosque first, then the two Rak'ahs of Fajr.