The original batch box was a 12" duct with the 8" duct connecting and going up through the riser. I think batch box may not be the right word. I think it was just not angled up enough and I found I didn't need that big a burn chamber anyway. When I converted it I just put a 8" elbow inside the 12" chamber and just filled it in with clay and perlite.
No left over charcoal after burns. The core is perlite with enough clay to make it stick together well. The exhaust going from the core to the bell does indeed get hot but not dangerously hot. Still, I plan to insulate it with rock wool, but it just adds to the radiated heat. I don't have cad but I have attached a drawing of the system. Sorry it's crude. I'm at work.
And yes that's a badass stainless lid I got off Ebay and I just keep a gap open at the bottom so that air comes in and cools the lid on its way to the fire. And yes it's just a coincidence that vent happens to be there.