Does uttering part of a supplication or "Rabbana wa lakal Hamd" (Our Lord, to You be all praise) after rising from prostration or bowing invalidate the prayer, and what if that occurs in the recommended (Sunnah) prayers?

Completing the rest of the supplication after rising from prostration is contrary to what is legislated. If it leads to abandoning the takbir of transition in its proper place, then the prayer may become invalid according to some scholars. As for descending for prostration at the end of the tahmid, "Rabbana laka al-hamd" (Our Lord, to You be all praise), it does not cause any harm.