What is the ruling on the prayer of someone who prayed as a follower behind someone other than the successor (al-mustakhlaf), and the ruling on the congregants transferring their intentions to the successor in the same rak'ah, and what is the ruling on the prayer of the one who was not made successor, and what is the ruling on my two prayers after catching two rak'ahs with the successor?

The prayers of everyone are, God willing, valid. It appears that the person who said takbir to the people intended to inform them about standing (for prayer), not about leading (the prayer). If he said takbir and raised (from bowing) with the people after the caliph (Imam) had already raised, then there is no problem. Otherwise, his ruling is the same as someone who precedes the Imam to a rukn (pillar), and his prayer is not invalidated according to the majority of scholars. There is no blame on those who raised (from bowing) at his takbir, assuming he was their Imam, and their prayers are valid because they are excused. Your returning to follow the mustakhlaf (deputy Imam) is correct, and by it you attain the virtue of congregational prayer.