Great job!!! My dh & I just finished our 1st and only coop - Most of our materials were reused/repurposed items. Once we painted the whole structure, the different plywood/siding all blended & didn't look as piece mealed.
As far as the plastic storage, I think you said it was 8'x8' - and you can't get the top on. Here is an idea. Leave the top off and the doors. Lay the shed on the ground with the door side open on the ground. With the top off, attach that side to the coop as the opening from your run into the coop. Does that make sense?
As far as attaching it to your wire/posts - I think you can drill holes in the plastic & use screws and washers to attach it to the wire and/or posts. Perhaps you could build a frame around the shed and attach the wire to the frame and the frame to the shed.
I'm not sure what to tell you about predator proofing the shed though (around the edges on the ground).
Have fun!