Do you know how deep he buried the tree roots? I suspect those places may not support grass growth for 5-10 years because of the massive amount of wood underground. There's really nothing you can put there, either. I might have suggested a gazebo or something that doesn't grow, but the ground will sink as the wood decomposes, so it can't be anything with permanancy. Maybe picnic benches and mulch? The best thing to do there would be to have someone come back to rip out the roots and resurface the area. That would all be done with the same tool - the trusty box blade on the back of a tractor. I'm not sure what your budget is but the box blade with tractor is one of the most economical things you can do in a hurry. If you did not have so many trees, it would take about 2 hours to do the entire yard. With the trees it might take all day. If someone suggests using a Bobcat or Skidsteer, those machines might take a full week to do the job.

The problem with burying wood is that it requires a special fungus to decompose. That fungus requires nitrogen from somewhere to live and do its job. Normally it lives above ground on fallen trees and will take nitrogen from the air. When you bury the wood, the nitrogen must come from the soil. It "robs" nitrogen from the soil preventing the plants from getting any. Furthermore, cutting off the nitrogen greatly extends the time it takes to decompose. I had a tree trunk buried under compost for a couple years. When I unburied it the trunk looked good as new. Hard as a rock. I tried uncovering it and keeping it wet. In 18 months it is weaker than balsa wood and extremely soft to step on. Wood absolutely has to be above the ground to decompose.

Spring is a terrible time of year to seed grass, but if you have to you have to. Don't spend a lot of money on any seed you buy now. If you are going to buy anything premium, get that for fall seeding. The grass you plant now will be hard pressed to live through the summer heat. Grass planted in the fall will be more than ready for summer heat next year.

One thing you can do to help yourself is raise your mower from the lowest setting to the highest setting. If that is Kentucky bluegrass, it will start spreading. When you see that, drop the mower one notch from the top setting.

Great pictures by the way. Is that rock much higher than the base of the trees? It looks about a foot higher.