Is praying two Rak'ahs immediately upon entering the mosque before Fajr, then supplicating and reciting Qur'an until the iqamah for prayer, a correct action?

May Allah bless you for your diligence in observing Fajr prayer in the mosque. Continue to perform all other prayers on time and in congregation. The Prophet, peace and blessings be upon him, said: "Whoever prays Fajr is under the protection of Allah." What you do, praying two rak'ahs after the adhan and continuing with recitation and supplication until the iqamah, is a correct action and a great act of worship. Whoever enters the mosque after the Fajr adhan and performs the Sunnah of Fajr, it suffices him for the "Tahiyyat al-Masjid" (mosque greeting prayer). The best thing to engage in between the adhan and the iqamah is recitation (of Quran), supplication (dua), and remembrance (dhikr).