I'd definitely NOT attempt deep litter in the coop with this flooring as the floor is already taking moisture damage (run is different, absolutely try deep litter). Deep litter favors having moisture for composting down poop, therefore it doesn't work in most coop set ups.
You mentioned trying sand for the coop, which can work if you're willing to scoop it out daily. I know there's specific sizes of sand that folks favor but I know very little about it, so unfortunately I can't make recommendations.
A sloped top/cover on top of the nest box will prevent birds from sleeping up there. A simple way to make one is to hinge a piece of scrap plywood over the top of the nests. 45 degree slope as shown below is ideal, I've watched a chicken try to sit on top of my nests and slooowly slide right off.
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I'm also curious as to how much roost space you have, as 23 birds = 23' in roost, and not sure you have that much, which might be contributing to birds sleeping on top of the nests.
I do have screens on my windows, just keep in mind they need to be cleared of dust every so often so they don't clog up (I use a broom to loosen and leaf blower to blow them out).
Incidentally chicken wire isn't predator proof, but it'd be costly to replace/layer on that much hardware cloth. Might want to look into electric for extra security against bigger predators.