Is it permissible to repeat the congregational obligatory prayer with family members at home to help them earn the reward of congregational prayer, and is the second prayer considered obligatory or supererogatory?

It is permissible for you to lead your family in congregational prayer if you have already performed your obligatory prayer in the mosque. This is based on the established fact that Mu'adh used to pray with the Prophet, peace and blessings be upon him, and then return to lead his people in prayer. Your prayer with them will be a supererogatory prayer (nafilah) for you, and you will be rewarded for guiding to good, as stated in the hadith where the Prophet, peace and blessings be upon him, said: "Who will bestow charity upon this one by praying with him?"