What does SOP mean?
We are some of the few carriage drivers who actually leave the reins attached to the bits. We find it easier, especially with the Liverpool bits where the slots are as thick as the reins, and I don't want to pull on my horse's mouth when attaching them. Yes, we have to run them through both the neck and the saddle terrets and buckle them at the end. Generally, I throw the reins over my shoulder or over the stall door (we have 4' tall mini doors, the horses are tied in front of them to harness) while I am bridling and getting the other rein threaded. I know that there are other drivers who always detach them. This works for us, and we have not had anyone (including Int'l level drivers) who have said that it is downright wrong.
As far as not moving while mounting the carriage, that really has to be taught before you ever get as far as getting in the cart. I use Morgan/driving trainer Jeff Morse's technique/philosophy that horses are poor managers and they don't get to make decisions. Jeff and I talked about my gelding this spring, and he suggested that I make sure that he doesn't move during brushing, harnessing, putting to, or entering the cart. To do that, you not only need to stop the movement, but put the horse back where you put him in the first place. He doesn't get to make the decision about where to go, you do, and you put him where you did, so that is where he needs to be. If you are consistent and calm about that, he will eventually figure out that it is easier to stay put, than to be put back into place. Also, take your time brushing, harnessing, and putting to, so that YOU are not in a big rush. The horses pick up on this. And then once you are in the cart, take your time before asking them to move off. Take a big breath. Enjoy the fresh air. Make it a "matter of fact" event. My gelding is a different horse after just being consistent about standing during harnessing! He stands great, he doesn't argue with me about other things ("I don't want to go by that!", etc.), he stops when asked, he knows that Mom makes the decisions, and it wasn't even a battle to train!!!