Is it permissible for a Muslim man to pray with his wife at home after praying in congregation at the mosque, and what kind of prayer would this be (obligatory or supererogatory), and is it necessary to call the iqamah for it, and what is the ruling on this for the Fajr and Asr prayers?

This act is impermissible for two reasons: Women are not among those for whom congregational prayer is obligatory, and the regular performance of a supererogatory prayer after every obligatory prayer, with a number of units equal to the obligatory prayer, has no supporting evidence and falls under the category of innovation (bid'ah).