Is the prayer of the Imam who entered while the people were performing the second rak'ah after he had given them permission to pray without him, then he pulled the Imam and prayed with them one rak'ah for himself and one rak'ah for the people, valid?

First: The resident Imam is to be awaited, as long as it does not cause hardship to the congregants. It is permissible to pray before him with his permission. Second: If the resident Imam arrives during a prayer led by someone else (whether with his permission or not), he may step forward to complete the prayer with them, building upon the prayer of his deputy, and the deputy then becomes a congregant. This is permissible and valid, but it is more preferable to let the deputy complete the prayer to avoid dispute. As for the prayer mentioned in the question, it is valid.