No, I don't think plywood is the norm. Of the runs I've seen that have a floor, it is usually a concrete slab or pavers. Any wood that comes in contact with the ground is going to end up rotting, so I'd be reluctant to do that.
You may find that the deep litter does the job. From what I've...
I can't tell by the pictures, is your entire run covered or just 1/2? If it isn't entirely covered, doing so will likely help things. I'd also look at where the gutter downspout from the house dumps and make sure that it is directed far away from the run area. You could also pitch the roof on...