What is the ruling on a latecomer completing his prayer with another congregation when one of them takes the place of the Imam, while observing the rulings regarding the Imamate of latecomers, walking to organize the rows, and changing intention?

What you did, a follower imitating a follower, is a matter of dispute among scholars. Sheikh Ibn Uthaymeen validated the prayer in this situation, but it is better to repeat it to avoid the disagreement of those who deem it invalid. As for the extra movements behind your second Imam, they do not invalidate the prayer if done out of forgetfulness or ignorance. However, intentionally adding movements does invalidate the prayer.