Once he gets older, running should be fine. Jackson runs for a while -- he went up to 1 mile but I've never tried further, mainly because I can't run longer!
I think small dogs can definitely run long distances and it's good for them, but yours is so little and so small, it will probably be too much. I got Jackson at 9 weeks old but he was larger at 5lbs at the time. I also, under my vets permission, took him outside because we immediately began outdoor potty training. He never went to a dog park or a Petsmart or a public park though until later, just my backyard and my dad's backyard with his 2 dogs who I know are healthy and vaccinated.