How can we reconcile the Prophet’s (peace be upon him) praying the two rak’ahs of Fajr (Sunnah) in his home and then sleeping until the establishment of the prayer, with the virtue of going early to the mosque and waiting for the prayer there?

It is commendable to go early to the mosque and await the prayer due to their great virtues. The prostration on the right side after the two rak'ahs of Fajr, as reported from the Prophet, peace and blessings be upon him, can be interpreted in several ways. One interpretation is that waiting at home is like waiting in the mosque, or that the prostration is specific to the Imam, not the congregant. It appears that arriving early for prayer is more emphasized for the congregant than for the Imam, as the Prophet, peace and blessings be upon him, usually came out for prayer at the time of the Iqamah.