The coop appears more than 4 foot wide. If interior space is a problem, you could remove the nesting boxes, add another roosting bar, and find an external egg nest solution. I have one of these on the outside of the henhouse:
View attachment 4164572
Regarding shade, the more the better. As others stated above, I would at least cover the entire run, withstanding the fact that there is already tree cover. Shade cloth is the best bang-for-your-buck, relatively maintenance free thing you can do to keep the birds cooler. Ideally the cloth would cover your entire setup, not simply draped over the arch cross-members but instead hang horizontal. Aesthetics aside, this can be easily accomplished by fixing tall posts to the existing vertical fence posts, fixing the cloth/cloths to the extended uprights.
I would not enclose the area under the house, at least not at first. I'm in a hot environment and my elevated hen house sits underneath a large mesquite tree. The hen house roof blocks 100% of visible sun. The chickens park underneath the hen house between about 11a and 2pm, when another spot becomes more favorable.
When you have free time, I'd replace the zip-ties with wire.
I do like the design of the house...it has a real 1970's feel.