Would you be willing to hire a guy with a tractor and box blade to regrade and leave you with a perfect surface for new sod or seed? Assuming he uses a real tractor he could/should be done in a morning. If he uses something like a skid steer or a bobcat it could take a week. You decide - they get paid by the hour.

What kind of grass do you want to have eventually?

I am confused by what you said about the water running beside your house. From the two pictures it looks like the slope goes from (Picture wise) from left to right, but also falls away from the house toward the street (Foreground). If this is the case, the creation of a swale to direct the water further out into the yard on the left side would take care of much of the problem. While I can not see the left side of the house you may need a little work to divert the water to the front of the house. While there does not look like much of a slope on the right side of the house with the swale and what slope there is could resolve this area.

To me the biggest problem with the front yard is the sidewalk? that come out towards the street. That could be rebuilt or removed without much work.

As for what appears as unlevelness, that would be best determined when you are on the mower. To me it looks like most of he problems could be resolved with a mattock, knocking down the few high spots (https://en.wikipedia.org/wiki/Mattock)

We live about 20 miles east of Raleigh, and have been experiencing 100 degree days with very little rain, but extremely high humidity. So, my recommendation would be you do nothing until some time in October. Then spending a couple days in the yard. On the first day cut and reshape the right side of the yard. Use the resulting dirt to fill in the area of the sidewalk. Next day walk around the yard knocking down the high spots with the mattock. The above is exactly what I have done in my yard and it resolved my problem.

The swale does not have to be deep, a 6 to 8" depression in the 3 to 4' wide and about 5 to 7 ' long. If you use what you remove to create a berm on the edge of the swale, it does not have to be dug out that deep. If there is good grass in the area first remove the grass dig down to the depth you want and replace the grass. It has to be wide enough that it does not create a problem for your current or future mowers

From the looks of the photos I can see there is a lot of shade. I think that is one of your biggest issues. If you have a southern grass chances are the shade is hurting it the most. It just looks neglected. Depending on the type of grass you have I would aerate and top-dress with compost (fall for cool season and spring for warm season). I would remove trees or prune branches to allow more light. You may want to pay someone to aerate and top-dress, unless you want to rent the aerator and do it your self. You will need about 1-2 cubic yards of compost per 1000 sqft to do a good top dressing. If it s cool season chances are you need to seed it as well during the renovation.

Thanks very much. Shade is definitely an issue. Perhaps light greenery would do better than grass? The little walkway mentioned is now complete (gravel) and looks much better than in the old photos above. I will definitely consider your advice, guys. Hopefully I'll have new photos sometime soon.

With those are gravel side walks you have take a big step in the control of water around your house. Work the yard contours so the water come down the slope through the gravel sidewalk. This includes the side walk that runs parallel to the house