Is it permissible for a Muslim to pray with his father at home if the father is unable to go to the mosque? And is it permissible to repeat the prayer at home after performing it in the mosque?

If the father is unable to go to the mosque except with difficulty, then he is excused from attending congregational prayers. As for the son, he is not excused if he hears the call to prayer. The son repeating the prayer with his father at home is considered a form of repeating the prayer in congregation, which is recommended by some scholars, such as the Shafi'is and Hanbalis. This is considered the prayer of someone performing an obligatory prayer behind someone performing a supererogatory prayer, and this is permissible according to the majority of scholars.