Save your money on weed n feed products. If you need a fertilizer, buy a fertilizer. If you need an herbicide, spot spray with something like Weed b Gon.

For dog pee spots you can scatter a heaping handful of table sugar on the spots as soon as practical when you know where the spot is or is going to be. The sugar stimulates the bacteria in the soil to process the urine and make fertilizer out of it. Quite often the dead grass comes back from the dead and starts growing faster than the surrounding grass.

As to seeding now, it depends on where you live. I'm guessing high altitude. That might give you a chance. Generally now is not a good time to seed because the summer heat is right around the corner, but it looks dadgummed cold where you live.

I just saw the Utah in your post. With that in mind I'm going to say you could overseed now with little problem. Watch for summer annual weed seeds germinating, too. When the grass is up to 4.5 inches tall you can mow it down to 4 inches and spray for weeds.