What about the architecture/design itself, is any of that open? Could I find a design, send it off to china, and have them print off a couple hundred, even at a much higher price than some other processor?

Uh, well I guess...? But generally speaking IC fabbing doesn't happen in the "couple hundreds". If you really want that, you'd be looking at MPW (multi project wafers), were a bunch of designs are slapped on a single mask to spread yield and reduce per-party costs. (It pretty much works like PCB group manufacturing - a bunch of designs is slapped together 'til the maximum size of the fab, and then that is produced a couple times, e-tested, cut and bad pieces discarded. Everyone gets a couple copies of their design.)

Today it's a bit better, since eg. GF can (in theory) offer recent nodes (eg 22 nm and smaller), which wasn't something you could get ten years ago (back then that'd be 65 nm - most low-volume and custom stuff has been iirc 130 or 180 nm).

Not quite because you have a few more steps in the process after the design phase. For instance, you need to perform testing on the fabricated parts to determine if there are physical defects. That means creating the tests then paying for the testing to be performed at the fabrication place.