I don't know if dogs dream or not, but I'm glad to hear that Rex's sleeping behavior isn't out of the ordinary; he does most of the stuff that you mentioned (except I don't ever remember him running in his sleep). He's very active and funny to watch when he sleeps; those times make-up for the times that he's an unholy terror when we're out on "pleasure" walks.
When he's awake, Rex is very strong and enthusiastic. With full-force, he lunges at things, people and other dogs; they act as though he is attacking them. I say: "ah, he loves everyone and thinks that everyone loves him too".
When he's active in his sleep, maybe he's dreaming about not being on a leash.