Is it obligatory to complete the supplication "Lord, forgive me" between the two prostrations after intending to repeat it, or should one prostrate with the Imam if he prostrates before completion? And is it permissible to say "My Lord, forgive me" or "Lord, forgive me," or are both permissible?

Remembering Allah between the two prostrations is a Sunnah according to the majority of scholars, and obligatory according to the Hanbalis. Saying "Rabb ighfir li" (Lord, forgive me) once is sufficient, though it is better to say it three times. If you say it once, you must then follow your Imam. The correct way to say it is without the "ya" in "Rabb" and without vocalizing the alif al-wasl in "ighfir" when connecting it.