Is it authentically reported from the Prophet, may God’s prayers and peace be upon him, that he would raise his hands and supplicate after the completion of the Adhan and before the Iqamah and before the opening Takbir (Takbirat al-Ihram)?

There is no basis for raising the hands and supplicating between the Iqamah and entering the prayer. This was not reported from the Messenger of Allah, peace and blessings be upon him. Rather, it should not be done, because it is contrary to the Sunnah.