I think Cooper will start, for what its worth. I think they really want Green to be the starter, and if healthy, he'd probably be an elite offensive guard, but he just can't stay healthy and never really has. They won't just cut him outright, because he's under contract and he can always maybe put it together health wise, but I wouldn't be surprised to see us go with a guard in the draft in 2018.
I agree on Claiborne, but not Carr. Carr was solid, if unspectacular, his entire time. He wasn't Revis with INTs, but he was a steady player his entire time here and with how the rules are for DBs these days, it's a miracle he was able to do as well as he did (or anyone who plays decent, for that matter - fantasy football officiating is basically set up for DBs to lose consistently). I also think the Cowboys made a mistake releasing Terence Newman several years ago, which I'll get into a little bit below.
I think Mayowa is a result of the same thing the Newman release and Claiborne drafting were a result of -- the front office, scouts, etc. giving too much deference to the defensive coach. All of these defensive linemen we keep drafting are Marinelli types -- maybe it's time the scouts and front office realize that Marinelli doesn't really seem to know what he's doing with respect to DL talent and just give him players and tell him to adjust. We've always gotten into trouble on defense when we try to draft and sign scheme fits instead of just drafting ballers and fitting the scheme around them like we're doing on offense now.
Edit: I also am enjoying this chat -- good football talk.