Is it permissible for a Muslim man to lead the women of his household in prayer after returning from congregational prayer at the mosque, considering it a supererogatory prayer for him? And in the four-rak'ah prayers, does he pray all four rak'ahs consecutively without making taslim between every two rak'ahs, and does he do what the Imam does in the obligatory prayer?

It is permissible to repeat a prayer that you have already performed with the congregation in order to lead others in it. This is the view held by Ash-Shafi'i, Ath-Thawri, Al-Awza'i, and Ibn Al-Mundhir. Their evidence is the hadith of Jabir, which states that Mu'adh ibn Jabal would pray with the Prophet, then return to lead his people in that same prayer. The Malikis disagreed with this, citing the hadith: "Indeed, the Imam was appointed to be followed, so do not differ from him." The first opinion is more evident because the hadith of Mu'adh is a clear text on the matter. The questioner should repeat the entire prayer, and there is no harm in performing it as a supererogatory prayer of four or three rak'ahs.