I would not build an elevated coop, build it to the ground so you can enter it. I have had the elevated one, and it was a royal pain in the ...neck. Difficult to clean, difficult to get at a hidden nest, and difficult to reposition roost if you need too.
Elevated coops do add a bit of space in the shade for a very small set up, but you have a lovely run planned.
As stated above, where you are will influence this, but I have a run built out of chain link fencing, and the bottom lined with chicken wire, and for the first time in years, it is predator proof. So for small set ups, the hardware cloth is and can be used, but for a larger area, I have been happy with the chainlink.
Do plan to make the run totally enclosed as in over the top if you have the possibility of coons.
Mrs K