My understanding is your lawn problem is part plant, part canine. Your dog's urine is burning your lawn, weakening it, and then killing it. (female dogs, in particular, cause this issue.) While your dog urinates apparently randomly, there probably is some sort of statistical pattern, as seen in the photos. :-)
If you didn't have a dog, and you had the patience, I would rake in fresh dirt into your bare spots, seed, rake, then water daily for 14 days. That will fill in your lawn, which is what I every year, so some extent.
But, because if your dogs, the seeding may not work as well. I would suggest sodding. go for the larger bare places first. place the sod (instructions all over internet) in open fresh dirt) then water the heck out of it for 7 days, cutting back the next week. this will make your lawn look much better. do this section by section, learning as you go. it wont be perfect, but it will look great.
You'll continue to have this issue, because of your dog. I would suggest working at in little by little. (sometimes, I'll rip out a piece of lawn, buy a piece of sod, and shove it in. I have a female dog too.) Also, and it maybe alittle OCD, but watering the lawn after the dog urinates will reduce the burning on the lawn. R
oh, consider keeping your lawn a little longer. I mean, ask your lawn guy to cut at a higher level.