Here is an Ideabook for a dog friendly yard. Click on the photo name to see the whole book. Another Ideabook for outdoor pet projects :)
https://www.houzz.com/magazine/8-backyard-ideas-to-delight-your-dog-stsetivw-vs~8824871
https://www.houzz.com/magazine/15-outdoor-pet-projects-youll-lap-up-stsetivw-vs~15509255

Here is a separate fenced off dog run area, click on the photo name to go to the comments section to read details

This mud yard sounds too familiar. If your tree is anything like ours at surface of soil you won't be able to loosen the soil prior to spreading grass seed bc of all the roots. As a diy project we placed about 1/2" soil on top of yard, dense shade grass seeds then placed hay over the seed to keep the birds away and too slow down rate of evaporation. Finally!!! This year we are getting grass. But, you will need to keep the dogs away somehow

Hey where did you get the picture of my back yard! LOL We have three dogs and have all but given up on having a proper back yard. For a while, we had a nice bed of leaves and pine straw that had created a natural "mat". Then the landscapers decided it would be a good idea to blow that all off! Left us with bare dirt that turned to mud when it rained and actually created a small erosion issue that we corrected. There is a fine mulch that you can put down but if you have dogs like ours like that like to run and dig, it's a constant challenge.

Thanks everyone!!!! Sounds like we are all in the same boat! I was thinking of putting rock boarders in. And then either seed or sod.

I just got new backyard grass this year. It is tall and dark green and gorgeous........and I now run around at all hours of the day and night, chasing the dog with the hose. My friend has a small backyard and they got synthetic grass with the rubber crumbles that look like dirt, but weigh it down. Her grass hoses off and never has brown spots. The dogs love it because it is warm. It always looks great. Maybe you can do more bushes and plants and leave a smaller area for artificial grass? I love real grass, but wish I had gone artificial in the back.