There is a lot of good information on line regarding grading around your foundation. Generally you want 10' out from the foundation to be 6" lower than the soil at the foundation. That gives you a 5 percent slope. You can go a little less but the closer to 5% the better you are.

After that 10' it's good to have the ground continue to slope away from the house so that you don't have a swale 10' out.

As far as pooling in the landscape, that's going to depend on your climate and how well water percolates through your soil. Increasing the organic matter by adding compost will help with that as well as improve your soil for planting grass. From the looks of things you don't seem to have any major depressions in your backyard. Are there any areas that you see pooling in currently?

I'll offer a tip that if you describe the situation in such a way that it sounds like a complex, detailed, trick algebra problem, you will lose participants. For a forum, the description should be simple.

There is no way to stop water from coming on to your property. What you need to make sure of is that there is no grading or "devices" added that RETAIN water and keep it from running off of your property. Likewise, you cannot concentrate the water and funnel it toward your lower neighbor's property. If the natural drainage is toward the west, that neighbor cannot install a board that dams the water to keep it from leaving your property. If there is such a board it would be to your advantage for it to rot away. Hopefully, the situation could be worked out in a better way rather than having to rely on the process of decomposition or calling Morgan & Morgan. Your photo does not show this board and how it is retaining the water, so it's not possible for us to evaluate. Please post pictures that show what's stopping water now and where the water should go. (No close-ups, please. It's also the surrounding context that's needed.) Though it helps to have sandy soil for drainage, you cannot rely on it as the only method for dealing with rain water.

rob, we'd like to know where you are, what zone. Care needs to be taken of the existing root system of your trees. Don't know your plan for incorporating the amendments you referenced. You can't till it in within the reach of your tree's root system as you'd be really messing with feeder roots.

Rosie

Forgot to add. Before you do anything to your soil send a sample to your local cooperative exchange office for a soil test. It will tell you what you need to add to it.

It is pretty odd to not have anything growing not even weeds. I see what looks like a stump in the middle. Were there more trees and shade before? If there was no reason for vegetation not to grow call your extension office and tell them the condition of soil and see if the recommend any specific tests.

As the stumps and roots rot they'll take nitrogen from the soil so you'll need to add extra fertilizer in those areas. I'd recommend using organic fertilizers because you don't have to really worry about over applying most of them and applying too much around the stump areas won't be a problem.

If your soil is as bad as you describe and your soil test shows your soil is low in organic matter (make sure you get that test done) I'd start by adding at least a 1/2" layer of compost over everything and work it down about 4" into your existing soil. 1" would be better and you don't have to worry about smothering any existing lawn so if you don't mind the money/work go for 1"

1/2" would be 1.75 yards of compost for your backyard. 1" would be 3.5 cubic yards.

If you're worried about hurting the tree roots with a rototiller you can use hand cultivators to work the compost in. At least around the trees. You can do the whole thing by hand. I've done a similar size with hand tools and broke it up into a few hours each day for 2 days and stocked up on sports drinks.

The compost will help with a lot of different problems including soil structure, pH, fertility, etc. With that much compost you won't even need to fertilize when you put seed down. But keep something on hand and sprinkle it over the areas where you cut down the trees once a month. You can use one of those hand held spreaders. I'd pick up some Espoma Plant Tone because you can get it in smaller bags than lawn fertilizer.

One final tip. As you work the soil and regrade it you're going to fluff it up and it will settle down once it rains. Contractors that do this work use various tools to get the soil to settle hopefully without over compacting it. Skid steer doesn't seem appropriate for your backyard but maybe a plate compactor? At the very least a lawn roller. Ask someone with more experience than this (maybe your local rental center?) I'm just providing this as a heads up.

Compost isn't a good medium to do the grading with so first take care of that, bring in top soil if needed or a mixture of sand and compost, then add compost where you plan on putting down grass.