ljrd3, For the first year treat these as potted plants untill the roots grow into the surrounding soil. The water must go directly into the potting soil that came with the plant. Use 12th of a gallon per day if the temperature is in the 70"s. Use 1/6th of a gallon per day if the temperature is in the 80's. Use 1/4 of a gallon per day if the temperature is in the 90's. Delay one day for each 1/4 inch of natural rain. The second year use the same total amounts, however, you can begin to water less frequently using proportionally more each time and also watering a greater area around the plants.

no i wouldnt mow it. i would plant a mix of native grasses and wildflowers. Generally its reccommend to use three or four species of grasses and about 10 to 12 species of wildflowers. they dont need mowing but can be weedwacked or burnt down once a year. we also use a mix of clovers and fescues on certain slopes as well...we reccommend that only be mowed, weedwacked or burnt once a year. the longer the height of the grasses the better the root system will be. this picture is stolen from metaxa another garden webber, but it represents the root system of fescue when left to grow to various heights. its apparent from the photo that the taller grasses have thicker deeper roots and to me its intuitive that if water is running down the slope the taller grasses with bigger roots will hold soil in place better than the shorter grass with little roots...

Grass only grows where there is sun. Where it will not grow because of lack of light, you must plant groundcover to hold the soil. Mulch can be used, too, but it is best as a temporary solution. Gravel mulch can be permanent, but its use is usually best limited because there is a downside to removing if future changes warrant. Also, it's more expensive. And it's not as good at preventing erosion as a good groundcover.. Not really sure what you are showing us a picture of ... the circular thing. Why do you think you need a retaining wall? I see plenty of room for a swing set.

Are the trees' roots exposed or are those mostly rocks? You want something that stays close to the ground to absorb the impact of the water. With shade your choices may be somewhat limited. Vinca is ubiquitous around here. You can plant other little plants in between. Various wild geraniums are popular around here, and liriope, perhaps some ferns. I had a plant called sweet woodruff covering the ground in the shade at my last place, very pretty in the spring. Some other classics are small bleeding hearts and ajuga, (actually anything in the mint family, like native mountain mint or horsemint which you can grow from seed) and perhaps some thymes although they need a bit more sun, also woodland strawberry and virginia creeper are natives that are used and there is a dwarf goat's beard that is beautiful. Yarrow is good in partly sunny areas. Someone on here once told me about creating little pockets of good soil surrounded by rocks. I see you are good with the idea of trying to spread that water from your drainpipes out as much as you can instead of concentrating it. A good thing to do is create pockets where the rainwater can be absorbed to control the erosion, but the trees roots run contrary to that objective. I would just try planting things and see what takes off. That's what I do on my hillsides, just keep trying to get vegetation established. We use natural wood chips too, to hold down the soil.