Is what is done in Fajr prayer—two adhans and an iqamah—Islamically correct? And when should the Sunnah of Fajr be prayed? And is it sufficient in place of Tahiyyat al-Masjid?

The Fajr supererogatory prayer (Sunnah) is observed after the second Adhan, which occurs at the break of true dawn. Performing it suffices for the greeting of the mosque (Tahiyat al-Masjid). The correct wording for the Adhan is that which was narrated in the Hadith of Abu Mahdhurah, which combined tarji’ (repetition of Shahadah quietly then loudly) with the quadrupling of the takbir (Allahu Akbar), and this is adopted by the Shafi’is. As for the muezzin’s saying: "Prayer is present, may Allah have mercy on you; prayer is present, may Allah have mercy on us and on you," it is not legislated and is an innovation (bid’ah).