Advice needed for predator proofing this

[Mama Bird 1960]

Arborist mulch should help elevate and keep their feet from being in mud all day -- however from your pics, the terrain looks to all be graded so water will be directed toward your chicken area - I'd assume wetter terrain will always be an issue with this location. The trees would definitely provide shade, but they're not helping with the protection aspect if they're interfering with making a secure run from those weasels, fishers and trash pandas.

Trees do provide shade but climbing predators can go from tree to tree and can climb down the trees inside your area unless there is a good roof above that to stop them. I created shade with structures in the run (like 2 pallets screwed together, standing up to form a triangle, and a plastic table they like sitting on and under.
Mama bird 1960

 

[Seaslug]

I also live in a wooded area with a stream and have a lot of wildlife around -- skunks, raccoons, bears, opossums, foxes, rats (from chicken farm down the road), mice, cooper's hawks & other birds of prey, including bald eagles, porcupines, snapping turtles... those are just things that I have seen on our property.
We've had chickens for more than 6 years now, and while we lost four early on to a mink that we did not know existed, it was due to a security lapse and has since been fixed.
These animals whose presence is know, however, are frequent visitors. And none of them have ever created a problem. The skunks & opossums regularly dig in the yard, the raccoons regularly are at the stream looking for food. Bears come through on rotation to see if bird feeders have been left out and foxes are about but wary due to domestic dogs being everywhere.
Some people will tell you that if these animals are near chickens your chickens are destined to be killed by them, and that they should be killed as a preventative for simply existing on your property near your birds.
I do not believe that, obviously. As long as there is habitat, there will be animals. The animals that are here currently are not causing me any trouble but if I eliminate them, it is possible that the next ones will cause trouble.
So my advice to you is that you provide your hens/birds with the most secure coop and run that you possibly can, as it will make co-existing with wildlife easier on all living things on your property. And know that, just because you choose to have hens/birds does not mean that you have to kill all other living things on your property.