Some coal does not burn down to a fine powder, it burns down to a hard crunchy ash, the pieces about the same size as the coal was.. When you have ash like this, you need to move the shaker mechanism enough that the pieces of ash get caught between the teeth of the grates and crushed by the teeth. Otherwise the ash will build up like you are experiencing..
So from what you describe,, you need to shake a bit more agressively.. to get the crunchy ash down through the grates..
You definitely need a Barometric damper installed.. this will retain more heat in the stove, and extend your burn time on a full load of coal.
With your stove in the basement,, you need to insulate the basement,, it looks like you have bare concrete walls in the photo, these will pull an amazing amount of heat from the room.. I'd put a layer of foam boards on the walls and insulate the floor joist ends and the sill plates..
I think you just need to learn your stove, the coal and your shaking technique some more..
The clinker is from a very hot fire.. if you have 350* measured on the surface [magnetic thermometer] of your flue,, you have at least 550* inside,, probably more like 600-800*.. You are burning your stove quite hot... I'd insulate the basement or move it upstairs..