Is the saying of some worshipers, "Our Lord, and to You be all praise and thanks," aloud after the Imam says, "Allah hears the one who praises Him," considered an innovation (bid'ah)?

We understand from your question that you are referring to what the jurists call "at-tableegh" (broadcasting/announcing). It is the Imam's role to vocalize the takbirs (Allahu Akbar) to announce the start of the prayer and the transitions within it. If his voice does not reach those behind him, then one of the congregants should announce on his behalf. However, if his voice is audible, then there is no need for such an announcement; at the very least, it is disliked. We also note that we have not found any basis for adding the word "wal-shukr" (and gratitude) to the dhikr (definition) (remembrance) recited after rising from bowing.