I saved the leftover pieces from my roofing material to use for wind block panels in the cold months. I only had 4 pieces, so for the other areas, I use plastic shower curtains.
These go up in the fall, on the west (prevailing wind) and north sides of the run. It makes it much nicer in there...
If I ever need to build another run, I will definitely be sure it is covered. I am SO glad I put a roof over mine. Avian Influenza is in my area, and my run is safe from wild bird poop, a vector of AI.
Absolutely make it tall enough for you to stand up in. Your back will thank you. And probably...