The back end space is probably fine. What I said about the tee was basically swapping the horizontal and vertical pipes, so the horizontal has the full-length smooth bottom. But I wouldn't worry about changing it unless you decide to rebuild the whole thing. There is no reason to have the horizontal extend back further than the vertical, that was just referring to the assembly method you described.
You will get a lot of combustion air in between the sticks in the feed, unless you have smooth straight pieces of lumber stacked tightly together, so the bottom air supply is just an additional kicker. The more common problem is having too much air beyond what is needed for complete combustion, cooling the combustion zone and reducing the effectiveness of combustion.
There is a concept known as "the three T's", or Time, Temperature and Turbulence. The gases released from the fuel by the fire need to stay hot enough to burn, for long enough, with enough turbulent mixing for all of the gases to meet oxygen. So the combustion zone needs to be insulated, long enough for the fire to complete before it reaches the top, and shaped so that the contents mix thoroughly.
As for clay, you can use potter's clay if you have an easy source of that, but most any clayey subsoil dug out of your backyard will work. If it will hold its shape when worked with enough water to make a stiff mud, and have enough strength after drying to poke or handle without falling apart, it will probably work fine. You can get more detailed descriptions by searching for "how to identify clay soil".
Mason Jar Soil Test
Urban Farming: How to Determine Your Soil Type
HOW TO FIND AND DIG CLAY
I would not advise trying to line the metal pipes with clay, but wrap the clay all around them. Making cob (clay mixed with short pieces of straw - dried grass cuttings work beautifully for this application) will make it easier to wrap the pipes. You can mix some perlite or vermiculite with the cob to improve the insulation, but the inner surface wants to be clay-rich for dried strength. Then surround this core with plenty of insulation.