Your dog seems to have allergy symptoms. I would try a few weeks on a fish based diet or turkey and similar treats. One of the food ingredients could be a problem. It would be easy to elevate the food bowl. Plastic can be an allergen too so I would try a ceramic one. Then if you wanted to you could add some of the homeopathic DERMA-IONX to the food or water to help reduce the allergy symptoms. I would also give probiotics and Omega 3,6,9 to improve digestion and reduce inflammation.
It is possible to be allergic to the carpet or shampoo residue. Maybe you can restrict him to a wipeable floor for a while to see if there is a positive change. Some dogs are allergic to grass. You could wipe the feet with a damp cloth when he comes inside after being on the grass.
Please keep us updated on your progress.