Seems more likely that the foot of clay soil is what's making your new lawn not very healthy. Clay without organic matter mixed in is not great for growing anything. It holds water and doesn't allow the water to sink into the ground, meaning the roots on plants and lawn are shallow. The old lawn is buried too deep below to matter. But even if it were not, it would probably contain good organic material for growing. In fact, some people turn over their lawn and just leave it in place for purposes of planting a garden, hedges, whatever. (If I understand what you said correctly you now have your old lawn buried under 1 foot of soil on top of which you've planted new lawn.)

We live in a town that has "soil" that's called sugar sand. If you look at it, it's a gray sand with a little dirt. Growing grass is hard, but after being here 7 years, I've managed to. We've actually sunk a lot of money into real soil over the years; but will people realize it? Probably not since people come here from town with real soil.. There's an agent that likes to photo shop grass. I always know when one of her listings come through my email.. it's that photo shop colored grass. Deceptive? To me it is, especially since I worked my butt off growing grass. One said "highway barely visible" that would only be the case if you never looked out your window or into the back yard, AKA the highway. You could not avoid seeing the highway unless you drive up and enter the house with your eyes shut. In fact if someone from the highway tossed their gum it would hit the roof. Are you looking in NJ at some of the new communities we've walked? Orleans has a really nice community in South Jersey, very pricey for the area.. guess what you can not only see but hear? I have no clue how they sell them.. well, yeah, I do. They're nice houses but come on. Does the highway not matter?

Drainage is the first problem to solve since it requires re-shaping the earth and no paving or planting can occur until it's done. From the picture all that can be seen is a flat lawn, so there's no obvious suggestions to offer. I think you will need to have someone who knows about drainage come and take a look. I would not call a "landscaper" but someone who comprehends drainage like a landscape architect or drainage engineer. I don't think it's a given that you will need to (or can) involve other neighbors in the solution, but a person figuring out how to solve the problem will need to know what is happening with grade in the neighbors' yards in order to solve what needs to happen in yours. My earlier illustration is indicating how you might lay out a path and firepit area. It's not specifying, or suggesting what anything is made of, which becomes the task at hand. Seems like the firepit area could be resolved easily with some gravel material that PACKS DOWN HARD. I would vote against using gravel that is loose. (The gravel you show in your first photo looks like loose gravel so I could imagine it causing trouble from the dog scattering it and trouble just from walking on it.) Actually, I could see a modified install of artificial turf as a possible path material, since it's flexible. I say modified, because the the standard install is almost like a brick patio. It requires a base of tamped granular material. The tree roots are going to be a problem so you may need to do something in an unconventional way, which of course, limits surety about how well it will perform. That's just how it is when "experimenting." One truth is that NO plant will grow in conditions that are not suitable for it. Therefore, you will NEVER have luck with Zoysia, perennial peanut or anything else that requires substantial amounts of sun. You must change the light conditions by altering the tree, or you must change plant material to something that will tolerate low light. It will never be any different than that. Actually, only one grass comes to mind that I know will tolerate the conditions and that is the wild native basket grass which grows all over Florida. FYI, it's invasive, but sometimes that's what one must tolerate when conditions demand. (See a picture of it here.) If you're anywhere near Brandon I can tell you where you can see it first hand, used in a public setting. It was not installed. It just took over because that's the only thing that would grow well. And it looks very nice mowed, too. It dies down to nothing in winter. Now or very soon would be the time of year to collect its seeds.

Neither of those is really a good choice but if that is all the choice you have use the least expensive. What is your base soil? Putting a soil mix that drains very well on a clay base that does not can present problems not easily cure. As you have indicated the 50/50 soil/compost mix will shrink as the Soil Food Web digests that compost and your level lawn will no longer be level. A good soil will have about 5 to 8 percent organic matter in it which might be what that screened "topsoil" may have. Keep in miond that "topsoil" is quite simply the top 4 to 12 inches of soi from someplace and it might be something worth the price or it may not be worth anything.