I'm in ohio so tends to get pretty wet
As ejcrist said, you want the coop dry. Wet bedding is not something you want.
As for the run, you don't have to give it a roof, but if you don't, the ground will be however the ground is when you get rain, stomped on and kicked up by little chicken feet, otherwise known as mud. LOL Also, it depends on how big the coop is and how much other shelter you've got. You could have an open run, with some little "huts" in there for the chickens to shelter under when it rains. Also, if you get a lot of rain, consider the site where you are putting the run, you want drainage away from it, and you don't want it in the path of water draining from elsewhere (so, don't put the run downhill of the coop roof, for example).
You're coming into spring and summer aren't you? Is your rain in winter? You could defer adding a roof for six months or so, if you are on a budget. You could also have part of the run with a roof, and part open. If it's hot there, you should have some sort of shade/shelter/roof.
Height of the run is for us, rather than the chooks. It is a pain in the neck having to crouch (or crawl) into it and you'll get sick of it very quickly. Full height is MUCH better for people. The chooks are fine with chook height, you could make it chook height if you wanted.