Hey now Gary, let's not be too negative. Sod is probably not going to work out for you since it can be pricey, but not sure what units you are using when you say 20X30, feet or meters? If it's feet, then that's not too big, and the cost might be less than you think, especially if you do the labor of laying it. You need to call some sod farms to get a ballpark price, that's your job. If you decide to seed, then the dogs are going to have to stay off the area for at least 6 weeks, 8 would be better. Maybe split the yard with a temporary fence and do half now, the other half in late summer. Ideally the seed bed should be free of weeds, and graded. You can do this yourself, to a degree, but a professional with a box blade is going to do a much better job (this would also be the case if you sod). For seed, I would either go with a Kentucky bluegrass blend that uses traffic tolerant cultivars that do well in your area i.e. ones with longer growing seasons like Rush, Four Seasons, etc. The other option would be to use Poa Supina. This is a European grass found in the Alps that has a long growing season, spreads aggressively, tolerates shade, does very well in traffic, and in some instances does better if there is traffic. The seed is pricey (35 bucks a pound), and you will need 2-3 lbs per 1000 square feet. Which ever grass you choose, you will have to water in extended periods of no rain, and you will have to fertilize.

I suggest you find the identity of those trees. Then prune according to their species. There is a Name That Plant forum on GW. As to the fallen leaves, you will need a rake and a broom. I do some of it with my hands. (Or hire someone) The way to remove dead plants is with a shovel. Water the area ahead of time so the soil is easier to dig. If there is a stump that is too difficult to remove just leave it. Time will take care of it. You have so much vacant concrete. It would be nice if you had patio furniture or really anything to make it less oppressive. It would help if we knew your USDA zone and location. If you are in California we have our own forum.

You can’t fight physics. If the dogs use this exact path every time they go out, you probably aren’t going to be able to get anything to grow there. I’d lay flagstones. Yes, you may just move the dirt spot further away, but I don’t see any other solution. Just get the spot out from the house and hopefully the dogs will “scatter” a bit and not follow a direct line to their tree.

I like it! I've also always liked the look of natural wood fence, but when I had a tall long fence installed a couple years ago we went with a dark brown color and I've been pleasantly surprised how it seems to recede into the background more and disappear. It makes the space feel larger also. I think you will like it :)