I deprimed exactly one lot of 50 rounds before I tumbled it in corn cob media. The result was a whole bunch of corn cob plugged flash holes.
I leave the primers in now, and size/deprime clean brass, which makes a lot more sense to me. Why run dirty brass through a die? Is pretty much what I figured. I suppose you could deprime, then tumble, then clean out the primer pocket, but that seems to add a couple of steps to the whole process and add time. IMHO, it's faster and easier to just tumble the cases then deprime/size.
As far as media goes, all I use is corn cob. No particular reason other than it's a bit softer when it spills on the floor, and seems to make less noise when getting sucked into the Kirby.
It seems that no matter how much newspaper I spread, and how careful I am, those little crumbs end up on the floor somehow.