Lowering the lawn

[jaotguy]

4" tall x 6" wide cement berms poured on the edges between the cement and lawn ....

 

[davesl708]

Landscape edging will keep the mud off the drive. Cheaper and easier than lowering the ground or raising the concrete.
As to drainage, the water will need someplace to go. Tile drains and gravel work well. Find the low spot and install a French drain to redirect the water.

 

[Gary Fowler]

From your photo, just lowering the soil isn't going to keep the run off from crossing the sidewalk, too much of an elevation change up to the trees. Water is going to run down hill, gather up at the sidewalk till it overflows and still do the same thing. You need some drainage ditches to direct the runoff to a spot where you can install a drain UNDER the sidewalk for the water to get off the lawn.

Raising the sidewalks by 4" would take about 22 yards of concrete (1800 sq feet of surface)but still might be cheaper than your planned removal of 4" of soil and re-sodding of the area which would be very labor intensive and still not solve your water overflow problem. The concrete elevation change would be at most a 2 day effort.

 

[JerryK]

Put in an winding elevated walkways/trail and sell tickets for nature walks.....

 

[in2toys]

You might start getting in to tree roots as well when you start digging

 

[kebo]

4" tall x 6" wide cement berms poured on the edges between the cement and lawn ....

I kind of like this idea too, but all of the low spots in the sidewalk will still hold water. Hard to tell if there is any slope to that yard, but that could exacerbate it, if he were to add the berms on the edges of the sidewalk. Also, it would make it hard to raise the deck and drive a mower across the walk, which I'm guess the OP may do sometimes just to avoid having to go around the long way.