I've come to rely on the pelleted bait - I often sprinkle it around new plantings of seeds to keep the slugs & snails from destroying the seedlings before they can take hold. I had a bad infestation of Cuban brown snails a couple of years ago & now they're almost completely gone from my yard. I buy it @ Home Depot - most garden supply stores should have it - it's iron phosphate & breaks down into fertilizer... Thinking those in the photo might be tree snails? This article says they are not a pest... Here is a link that might be useful: Tree snails

Take this diagram as a very general idea about where you might consider placing trees. It doesn't mean you can place trees at any and all locations along the line, but that it's a starting place to evaluate tree placement. Often one tries to keep some centrally located open lawn space, but this is not a hard rule. Much depends on your specific needs and desires. You would want to take into consideration the specific tree and its size and habit, other competing features in the yard, and architectural features of the house, etc. If there are any very small trees (in the 10' to 15' range) they might go closer to the building in front of blank walls. (For these type trees I usually use multi-trunk trees that are more commonly thought of as large shrubs.)

You can't add enough gravel or soil to flatten the slope enough to make a suitable badminton court without killing trees. If there are small holes, you can fill them to even out the ground a bit to make for easier mowing, but I wouldn't do more than that. It's clear that the ground is getting enough sun for a light layer of shade tolerant grass, but even if the ground was level, this type of grass isn't dense enough IME to stand up to the foot traffic that a sports area gets. If you want a badminton court, I would put it in a spot that is level and has enough sun to grow dense grass. This might require a low retaining wall with proper footings and the removal of more trees or siting elsewhere. The area you showed in the photos will be most easily maintained for now with mowing I think, though as the trees fill in branches to take advantage of the new light and space created by tree removal, it will not continue to support grass, so you may want to plant a groundcover - depending on the type of trees, you may have issues growing many plants since shallow-rooted trees compete for moisture and nutrients with herbaceous plants and shrubs. In my area there are some energetically spreading ferns that make a lovely ground cover in this type of area. At least if you are in the eastern half of North America, you will have weeds, particularly woody weeds (maples, honeysuckle, buckthorn, oaks, etc.) since this area always moves towards forest if left to its own. So to my mind, the woody plants seeding in is a far greater issue than removing leaves from gravel in the fall. Mowing will discourage woody plants from seeding in - that's how we returned our brushy fields back to fields and how we keep the woody plants from becoming reestablished both in the fields and along the sometimes shaded road verges. And you can't mow gravel, so weeds (woody and herbaceous) would need to be removed by hand if you spread gravel since even a weed whacker can throw gravel to a point of being dangerous. Gravel is a fine mulch if you live in a very dry area, but not good in areas with a respectable amount of rain IME.

Sorry for the delayed response. Really appreciate everyone's input and learning a lot here. I think my next step is to research everyone's advice as well as visit some big nurseries and see what recommendations they have and go from there. I'll try to ask my neighbors if they know any more about the hill (and how deep the soil is) and any advice they have to maintain it. @Cathy Anderson I never heard of that word. That's exactly what I need! @Justin Holbrook I will look into your suggestions. @L Sure I will check it out. @ken_adrian Adrian MI cold Z5- The wall is not that tall. It's about 3 feet so I just climb it. Thanks!