I looked at the brake kit on Corvette Centrals website part number 193121 and it looks pretty cut and dried to me. Make sure that is the kit number you got.
The rotors and hubs and calipers will get hot even after a short driving distance as the pads are always in contact with the rotors. Because of the friction they will be hot to the touch. Hot enough to burn your finger for sure. If there is a problem with a front caliper the car would pull to that side when you step on the brake. If a rear caliper were dragging you would see it on the pad surface. In reading your posts it appears that your problems started when you adjusted your brake cables, so I would think that the problem is somewhere in that area. I don't think it is your calipers. It might be a defective master cylinder but I personally think it has something to do with the emergency brakes. Now the only way to find out what is wrong is a process of elimination.
Heres what I would do. I would get the complete parking brake kit, either part number 192088 for stainless or 192164 for steel from Corvette Central if you like. The reason I would do this is you get all the stuff you need to overhaul the parking brake system, the springs, shoes, adjusters, cables everything. This would accomplish two things, one you probably will never have to worry about your parking brake system again, and two you eliminate these parts from the mix as being the cause rather than replacing one at a time.. The only componant left would be the rotors. I would look closely at your parking brake flanges (corvette central part number 582032) and make sure none of the holes are elongated or messed up. Those would be tough to replace as the spindal would have to be removed so I would inspect them very carefully and compare them to the pictires. By doing this you will have completely restored your emergancy brakes to new and eliminated them from the list of posibilities as the problem. Now when you install the brake shoes you want to apply a little white lithium brake grease to where the side of the shoe rides on the brake plate. Also you want to apply a small amount to the adjuster threads and the other end where it rotates on the stud to ensure they spin freely. USE THE GREASE SPARINGLY, and be sure you don't get it on the brake shoe surface. (an old trick) As I said these emergency brake systems are poor at best and can be a real problem. They use the same system on the C5's and they are a constant problem also. Once completed I would never use the emergency brake again.
A couple of questions about the P/B install. Did you bench bleed the master cylinder before you installed it ? You say you replaced a caliper. Did it have one bleeder screw or two. After installation did you bleed all four calipers starting with the furthest away from the master cyl, r/r, then l/r, then r/f, then l/f ?
From everything you posted I would say there is something wrong with the emergency system for sure. Corvette Central has a Tech line for help. Call them, explain the problem, what you did and so forth and see what they say.