Depending on how much clients budget is, I would think the right thing to do would be to install drainage at the bottom level area of yard. Use some type of ground cover to plant hillside. juinpers, contoneaster, ect. Planting this area would elimanate mowing and alot of maintenance.
Retainging wall in the back, maybe two sets with some stair and planting areas, pipe the drain spout out to somewhere else, you might want to check there gutters, might be clogged. Maybe get some large River Stone white colored or something and put that in instead of grass make it kinda like a walk way, put a couple inches of it in along the side of the house
The first thing that I would do would be to make sure that the grade at the house is up to the 8" below the siding where there are no transitions.
Then grade away from the hose at a minimum of 2% for a minimum distance of 8'.
Where there are transitions in elevation of the bottom of siding or a door like in the photo, I would retain it to make sure that the slope going away from the house is greater than that going along it.
I would get the downspouts piped away into drywells or daylight if possible.
Then I would address what is happening at the bottom of the slope that I put in that goes away from the house. Depending on where I can get that water to flow, either parallel or away from the house. If I can continue to have it flow away on the surface without causing problems, I would. If not, I would put in a catch basin and leaching pit.
This will all define what you have left from there to the property lines. Depending on the remaining slope, how the property is to be used, and the budget, you can grade it or build a wall.
Everything revolves around the fixed elevations that you can not control. They usually are the siding and doors on the house, a septic system, property lines, driveways, vegetated areas that you want to preserve, and areas that will accept water without causing problems. Always start with moving water away from the house and getting rid of it. That defines the lowest of your elevations. Then make everything work in support of that.
If you start by building a wall, you lock yourself to another fixed elevation to deal with when it comes time to deal with the water. It will limit you. The wall should be built to deal with what is left after your drainage design solution is planned out.
The upper side of the downspout area on the lawn.
cut a trench 4' out from the house to the yard. This is to act as a french drain. Not comming down the hill but across the hill.
The French drain should connect with the downspout drain 4-6
feet out from the house. T shape
Install a drain box under the down spout.
Come out from the house 4-6' and dig a trench,
Install drain tile, into the box. Run the tile to a desired drainage location and backfill with decorative gravel.
Against the house, you will now have room to install a mulch bed.
Inkberri holly would be a good canidate to plant in that bed.
Where possible the gravel drain should be at the outside edge of the mulch bed.