Oh my. In ten years those trees might provide enough shade but definitely not now. You are going to SOOO need massively more ventilation. The easiest first-resort tactic would be to make an all-hardwarecloth door (just a wood frame, diagonal brace, and the other 90% of the thing is just hardwarecloth. This could either have a bolt-on panel for wintertime, or just replace it with a solid door for winter. ("Just replace it" is glossing over some technical aggravations you would face, but it can certainly be done)
Even that will probably not be enough, though, so I think you are going to have to face the task of cutting holes in the metal siding of the building. That shed type is not one I am familiar with, are the walls each a single big panel or are they made of 2-3' wide corrugated panels that overlap (like normal metal roofing or siding panels). If the former, you will need to drill corner holes and connect 'em with a Sawzall w/appropriate blade in it and make lotsa sparks. (Don't sever any studs!). But if the walls are made of multiple separate panels, you could just remove the topmost panel from one side and put hardwarecloth on and a hinged panel to cover it, and have a nice biiiig vent running the whole length of one wall. That plus the door being mesh might be enough to keep it from becoming an oven.
If you won't do any of that, at least consider buying a BIG BIG piece of 90% shadecloth -- you will probably have to special order it from Farmtek or somewhere like that -- that is perhaps 2-3x the size of your shed's footprint. Put that over top of the shed, so it covers the roof and string it TIGHTLY out to the run fencing on all sides so that you have a great bit shadecloth 'roof overhang' all around. This will keep the sun mostly off the coop walls, which will keep indoor temps from soaring as high, as well as providing some extra outdoor shady area for the flock during the day.
The problem is, if the coop gets too hot during the day, you may not be able to safely lock the chickens in it at night without cooking them, leaving them outdoors and vulnerable to predators (it is FAR easier to THINK you've built a totally predatorproof run than it is to ACTUALLY build a totally predatorproof run)
Oh, btw, returning to the original subject of a window -- easiest thing is to put a window panel in the door, or (simpler) just use a door that already HAS a window in it
JMHO, good luck, have fun,
Pat