Is it permissible to supplicate with: "O Allah, forgive me by the intercession of Your Messenger Muhammad, peace and blessings be upon him; O Allah, forgive me by the prayer of Your Messenger Muhammad, peace and blessings be upon him," due to the established fact of the Prophet's intercession and his prayers for his Ummah?

The intercession of the Messenger of Allah, peace and blessings be upon him, is of two types: specific intercession for a particular person, in which case it is permissible for the one interceded for to say: "O Allah, grant him the right to intercede for me"; and general intercession, which will take place on the Day of Resurrection, and it is not permissible to use it as a means of supplication today. His supplication, peace and blessings be upon him, is also of two types: specific supplication for someone he supplicated for or interceded for, and in this case it is permissible for the one supplicated for to say: "O Allah, answer your Prophet's supplication for me"; and general supplication, upon which it is not permissible to build a means of supplication, because supplication is an act of worship based on divine revelation. Thus, the permissible means of seeking nearness (tawassul) are through one for whom the Prophet supplicated or interceded specifically, or through righteous deeds.