The idea with the dominant eye is to have it lined up directly over the top of the arrow. If it's not, it distorts the sight picture. The mind can compensate, but I think it is better not to have to do that.
Try this with your dominant eye. With both eyes open, point your finger at an object on the wall. Now close or cover your dominant eye. Your finger will appear to "move" sidnificantly from where it was pointed. Closing or covering the non-dominant eye does not have that affect.
As far as learning to pull with a different arm, I haven't tried it, but I think you are probably right, it is just a matter of learning with a different arm. But it sure would seem to be easier to pull a bow with the stronger of the two.