Is there a specific supplication to be recited after the prayer has been established, and is it permissible or an innovation for the worshipper to say, "We hear and obey, Your forgiveness, our Lord, and to You is the return," when the Imam says the opening takbeer?

Abu Dawud mentioned that Bilal began the Iqamah, and when he said, "Qad Qamat al-Salah (The prayer has begun)," the Prophet, peace and blessings be upon him, said, "May Allah establish it and perpetuate it." This Hadith was deemed weak by Al-Albani and Abd al-Qadir al-Arna'ut. As for saying "Sami'na wa Ata'na (We hear and obey)" upon entering Ihram: if you mean saying it before entering Ihram, then it is an innovation for which nothing has been authentically reported. If you mean saying it after entering Ihram, it has not been authentically reported in the opening supplications of the Prophet, peace and blessings be upon him, that he used to say it.