Answer: It is difficult to establish grass in a shady area so I'd vote for a ground cover instead. Some ground covers will tolerate foot traffic but none will stand up to what sounds like a dog run along your fence. You can try growing vinca minor. It's a vining ground cover, grows about 8" - 10" tall, flowers in the spring and summer and will tolerate some foot traffic. Anything lower growing will disappear after a few months of doggie traffic. Once the ground cover is established the soil erosion problem should disappear.
Best wishes with your landscape.
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