Dear Jon,

Thank you for contacting Ask Extension about pet-safe groundcover plants. I do not know of any plants that meet all of your criteria, but you can probably find some that work in various growing conditions. The biggest limitation is sun and shade since plants generally prefer one or the other, but rarely both. 

The problem of nitrogen burns from urine affects many plants. Some can withstand foot traffic a little better than others, but turf grasses are by far the most durable. Even with turf, many species will grow in sun but less well in shade. 

Here is a rather extensive list of plants, the first group being toxic to dogs, and the second group being non-toxic. Some may be considered "ground covers" but this is a term that can mean very low growers up to sub-shrubs. 
https://www.aspca.org/pet-care/animal-poison-control/dogs-plant-list

Another list that may be useful is:
https://www.purepetfood.com/blog/the-ultimate-list-of-dog-friendly-plants

It is actually easier to find lists of plants that are toxic, rather than non-toxic. For this reason, I would suggest you research any plant that you are thinking of planting. Your veterinarian might also be a source of information for you.

I'm sorry there isn't information that is more specific to your needs, but I hope some of this is helpful.

Hi, Lynne.  I just realized I never thanked you for your email response.  I apologize for that.  Thank you for the detailed response and articles, which I did find to be quite helpful.