Is it permissible for me to stand to the right of my friend and be led by him in the Dhuhr prayer, or should I let him complete his prayer and pray Dhuhr individually, after he had been led by me in a supererogatory prayer, thinking it was the Dhuhr prayer?

Your friend's prayer behind you is permissible and valid, because, according to the preponderant opinion, a difference in intention between the Imam and the worshipper does not affect the validity of the prayer. If you finish your supererogatory prayer and your friend stands up to complete the Dhuhr prayer, you must join him in congregational prayer, because congregational prayer is obligatory. The fact that he was previously your follower does not prevent this; for if it is permissible for an Imam to become a follower in a single prayer, it is even more so in two separate prayers.