Is it in accordance with the Sunnah or legally permissible for the Imam to face the congregants after Fajr prayer until most of them have departed, then turn to face the Qiblah and engage in the remembrance of Allah until the sun has risen to the height of a spear?

It is Sunnah for the Imam, after concluding the prayer with the Tasleem, to seek forgiveness three times and recite the prescribed remembrances (dhikr). Then, he should turn to face the congregants and quickly move away from the Qibla towards the worshippers. The wisdom behind this is to teach them what they need, or to inform newcomers that the prayer has ended, or to remove any sense of pride from the Imam. The act of facing the congregants ends when its purpose is fulfilled. As for the Imam sitting after the Fajr prayer until sunrise, this is Sunnah. It is permissible for him to face the Qibla after the congregants have departed, even if this was not explicitly narrated from the Prophet, peace and blessings be upon him.