To answer your first question, I usually like horses to swing in to face me when I lunge them and ask them for a stop. However, what I do not like to see at all is either the horse walking towards me when I ask for a stop, or the horse predicting a direction change. I would rather have the horse stop without turning in than have him do either of these things. Make sure that you do not always change directions when you stop your horse on the lunge, or he may come to anticipate this change and move off in the other direction before you ask.
As for your second question, it will actually be easier for your horse to canter on his "off" lead without the additional weight and challenge of supporting a rider. Of course, riding does give you more specific control than lunging, so you may prefer working this out under saddle; however allowing him to develop and become comfortable with this lead without balancing you at the same time will be easier for him. So yes, it is not only fine to let him canter on this lead on the lunge, but I would recommend doing so. In fact, on the ground you can encourage him forward a lot easier than in the saddle. When I want my horse to go more forward on the lunge (or in this case, to not break from the canter), I start with a harsh vocal tone, then raise my whip and bring it closer to him, then shake it, and if I still have no response I will then crack it. This way he has a chance to learn that if he responds to vocal cues, he can avoid the scary whip-cracking. Make sure that you don't have any pressure on the lunge line when you ask him to go forward, as this can cause mixed signals and confusion.
Finally, I posted a few days ago about the Pessoa system. What is my opinion? It's junk. It's not teaching your horse much of anything useful. You need to be able to teach your horse to round down and collect with your aids and with proper training and muscle development. This comes with time and correct use of aids and exercise. If you are having trouble with this, you can always ask a trainer to help you (although if your trainer is using this system, I would probably not stick with her for such advice, no offense). Devices such as the Pessoa only force your horse into a false frame, which not only does not help building in up muscle development and proper training, it also teaches him to fight pressure on his mouth or to overbend. Not only is there never a need for such a system, but its use is a crutch that only do harm in the long run.