I was thinking a bit about this one.

Combining 1, 4) Any Sun or Sun and Shade mix off the shelf would work fine. Pick something with no noxious weed seed and the lowest level of weed seed possible, which means it'll be more expensive than the cheapest by a few dollars, which is what you want. You'll also get slightly better grasses at the same time. I'd suggest a standard tri-mix of fescue, rye, and bluegrass. Initial fertilization isn't necessary, but covering the seed with some peat moss is helpful to keep it damp while it germinates.

2) No. If the soil is ready for seed, you're good to go, so if it's flat and you can see soil, you're fine.

3) Unfortunately, no. Seed can be put down for a great lawn in late August to September with watering. You can put down seed in February and hopefully get some sprouting in spring, but you'll need to coddle it over the summer--and still need to seed again in August and September because summer losses tend to be heavy. Or seed in spring, water, coddle through summer, and probably still need to seed in August or September because summer losses tend to be heavy on spring seedings...

5) Those will wear at the grass a lot. Yard ranging dogs wear at it less.