Is the regular Sunnah prayer sufficient for the prayer between the two calls to prayer, and if not, is it permissible to perform both prayers in the mosque, and which one should be started with, and is it permissible to perform the Sunnah prayer between the two calls to prayer at home?

If the Sunnah (definition) between the Adhan (definition) and the Iqamah (definition) is the prayer of "greeting the mosque" (Tahiyyat al-Masjid), then it is not intended for itself, and the regular Sunnah prayer of Fajr (ratibat al-Fajr) or other regular Sunnah prayers suffice for it. It is permissible to pray supererogatory (nafl) prayers at home, and that is better than doing so in the mosque. However, Tahiyyat al-Masjid is prayed only in the mosque.