The way I’d attach things to that plastic is to make a sandwich unless you can attach to the framing. Take two boards, maybe 1x3’s though you can use 2x4’s, and put them on each side of the plastic, then screw them together. You greatly stiffen the plastic and have a solid connection to attach things to. I’ve used this method on wire fences and metal buildings, I don’t use plastic, but the principle is the same. Drilling pilot holes can help a lot with screwing these together.
Here is an example of how I made a pop door through a fence using sandwiches. Obviously I was not going for beauty with this and I had 2x4’s extra so that’s what I used, but it’s very functional.
Terminology can be a problem because the terms can mean different things to different people. With some set-ups there isn’t much difference in coop and run, with many there is. It does lead to confusion.
For what it is worth, the floor of my coop and run is dirt. I use wood shavings in the coop as a bedding material, but under that is just dirt.
One fairly common way to stop critters from digging in is to use an apron. Basically you take a piece of mesh wire like hardware cloth or 2x4 welded wire fencing say 18” to 24” wide and lay that flat around your coop and run. Then attach it to the bottom of your coop and run so nothing can crawl in between. You don’t have to bury it, but probably need to weigh it down with something until the grass grows through to hold it down. I like to take up the turf and put that back on top. That keeps it out of the way of weed eaters and lawn mowers. The idea is that a digging critter goes up to the fence or wall and starts to dig. It hits the wire and does not know to back up. It’s extremely effective and especially if your soil is rocky, pretty easy to install.
What kind of foundation are you setting that building on? You’ll need to anchor it down so wind doesn’t blow it away. If you use something like treated wood that gives you something solid to anchor that apron to.
I don’t see where you are located so I don’t know your weather, but wherever you are ventilation is important. Heat kills a lot more chickens than cold. In hot weather you need enough ventilation to let heat escape, otherwise you could build an oven. In winter you need enough ventilation to let moisture out, even when it is well below freezing. The main danger in winter is not them freezing to death, it’s frostbite. If your coop is so tight the moisture from their breathing, poop, and maybe a heated waterer can’t escape but builds up, they are a lot more prone to frostbite than if they have good air exchange.
I don’t know what they plastic shed looks like. You may have to cut holes in it for a window and ventilation. Putting those sandwiches around those holes so you can firmly attach something to it will be a bit of work. There will be a learning curve when you start to do that, especially if you are not really handy, but you can do it. Since the shed is plastic, I’d consider building self-supporting nests and roosts, not supporting them on the plastic. Or, depending on what the framing of that shed looks like, maybe attach them directly to the framing and not go the sandwich route for supporting them.
That shed can probably make a really nice coop. The plastic may be a bit hard to work with, but a metal shed would run into many of the same issues as far as attaching things to it.