Try variegated vinca vine(a perennial). I have not found a place where it does not grow. It has the character of being invasive and forms a dense ground cover that keeps on growing until freezing temperatures persist. Under the conditions you mention invasiveness is to be desired. The variegated type is very attractive and this blue flowering creeper will brighten a shady spot with long flowering periods from spring till fall. If it does not rain heavily tomorrow I will take some photos of a section I planted two months ago to give you an idea of how fast this vine takes off. Otherwise I will post them in a few days. As far as it's compatibility with dog traffic, it will look like the dogs were flying over it. Nothing gets this plant down and as far as the other things dogs do-again-nothing gets this plant down.

I was going to suggest a fence, I can't think of another ground cover that will stand quite that much running on.

The previous statement brings to mind another advantage to vinca vine. This ground cover is like placing a chicken wire fence horizontally; cats and dogs don't like to walk on it because the vine forms twisted loops that catch their feet. It's tripped me up a couple of times walking on it. I surrounded one of my ponds with it to keep my neighbors cats from hunting at the ponds edge and it has done the job.