I don't recommend using side reins on any green broke horse, it ruins their mouths. On any green broke horse, it is going to be difficult to truly teach any form of collection or balance with side reins.
Side reins are not meant to force the head into frame. They are meant to allow the horse to reach for contact my stretching the neck, which may later result to being in frame. In very advanced training, they also help with self carriage. With that said, you want your side reins to be loose.
There are many trainers that will not allow their horses to be ridden with any contact during the walk. It is said that it can ruin the horse gait at the walk. I've never personally witnessed a walk gone sour by the use of contact, so I am not so sure how much I believe this theory.
The bouncy motion - I'm happy you questioned this. Let me ask you- if you were to ride your horse on loose reins while bouncing and jabbing the horse in the mouth, do you think he'd be less likely to accept contact from the bit in the future? In my opinion, side reins do not make soft mouths.
And as far as being even on both sides - I have seen people do both. I have seen owners tighten the inside rein to mimic some sort of flexion. 99 percent of the time the flexion is false because the horse is only bending through the neck and not through the body.