Sorry been real busy at work and haven't had the time to get on here as much as I'd like.
Anywho, you are describing the typical triburner setup used by the older Alaska stoves. Triburner means the 1 rheostat controls how fast the combustion air (under the coal) flows, controls the convection fan (blows over or around the stove into the room), and controls how fast the coal is fed onto the grate.
As was mentioned, separating out the combustion air is the first step. However on the Alaska stoves the same motor that controls the combustion air is also connected to the feed linkage. The only true way to separate out the combustion fan is to get another fan and attach it to the lower opening of the stoker mechanism. Then you have to remove the squirrel cage from the shaft of that single motor. This way the motor control would only operate the stoker part of the triburner. What some of the guys have done in the past was to rig up a bathroom fan with flexible hose going to the lower opening. Or you can experiment on your own as to the type of fan and methods to attach it to the opening. I just keep mine at full power and just adjust the cam lobe. Been like this for 20+ years and no real issues.
Having the carpet move 2" is wayyyyy too much. As you have found it should only move 1/2 to 3/4" at a time. The maximum amount of coal burning allows 1" of ash at the end of the grate before falling into the ash pan. On my old Alaska stoker, I move the cam lobe 4 turns from idle to full burn. That is all. Any more and I'm pushing burning coal off the grate. And if you turn the cam lobe in too much, the fire will go out. It is that simple.
Different coals will burn differently as you see. Some of the so called unburned coal is actually shale mixed in with the coal. Having a little mixed in is a good thing as it prevents clinkers from forming. Having too much is bad in that you will not get heat from it. and you just paid over $200/ton for crap. Some here weigh their coal before burning and then weigh the ash afterwards to check how well their coal is burning.
As far as your wall getting too hot, you can always put a temporary heat shield in there when burning hot. Or how about putting a small fan to blow air around the back of the stove area and out into the room ???
Any other questions, ask away.............