I see 5 nestboxes.
If the nestboxes are 1 foot wide, then the coop is 5 feet wide.
The other dimension looks narrower, so maybe 4 feet.
A coop 5 feet by 4 feet would be large enough for 5 standard-sized chickens, allowing 4 square feet per chicken.
(If you actually measure the coop, it may turn out to be a different size, so it could house a different number of chickens.)
If hardware cloth (welded wire, 1/2" holes) is not an option, I would make a run with welded wire with larger holes (like your friend's run) instead of chicken wire.
If the run has no roof, I do not think it matters whether the coon can reach through the wire, because it can just climb in the top.
If the run does have a roof, a coon can only reach a certain distance through the large holes of the wire (maybe a foot or two.) But the sturdy wires keep out the body of a coon, and all of a dog. Chicken wire can be ripped off to let the whole coon or the whole dog into the run.
I would use hardware cloth for any openings on the coop itself, and shut the chickens into the coop during the night, because chickens are much more vulnerable while they are sleeping.