What must one do who caught up with the imam in prostration without bowing, and mistakenly thought that this counted as a rak'ah, then discovered his error after some time, and forgot the number of prayers he missed?

You erred by counting the rak'ah whose bowing you did not catch with the Imam, and you did well by making up that prayer. As for the other prayers where you do not remember if you made the same mistake, nothing is required of you, and the default assumption is the validity of past prayers.