I had been going to wait and let the chickens enjoy destroying the grass before putting in the wood shavings or mulch - what's the difference between those? - but if you think it's better I can start in with that.
Don't use pine shavings in the run. Use either tree trimming chips from a tree company or undyed mulch, like cypress from Lowes. The only reason I would cover the grass right of way, is to keep the flies away. You should be ok for a couple weeks without covering the grass. Once you start seeing flies, I would get the chips or mulch down quick.
I do want that kind of bedding on top of the hdpe floor in the coop, right?
As well as the sweet pdz in the poop board tray?
I use pine shavings inside the coop and sprinkle a little PDZ in it also. If it starts to smell, I add some more shavings to freshen it up. I clean it out once a year or longer. I use the white pine shaving bags from TSC, it has less dust than their yellow bags.
Yes, PDZ in the poop board. I have it 1/2" or less deep. There is no need to have it any deeper. I made a board with eye hooks to level it out after I scoop the poop. I also have a wire office basket that I sift the pdz through once a week, this gets all the small pieces of poop out of the PDZ. Here's a pic of my poop board tools.