You are working WAY to much.
First of all, yes, I would run a dry patch down after the wet patch. You don't want to run the risk of getting the powder wet. And there will be some wet spots on the barrel after using #13
Second, TC#13 is good and I have used a lot of it, but I have found that spit really works just as well because 777 is water soluble. But I have no problem with using it. Have done so and will again probably, so no real issue there.
You are not shooting on a clean bore. With what you are doing you are cleaning the excess fouling out of the barrel, but it is still fouled, and I think that is ok. I run the wet patch down, both sides and then a dry patch and that is still considered a fouled bore. It is fouled with powder residue in all the little imperfections in it until you do a good cleaning back at the house with multiple passes with a brush and solvent soaked patches.
You are way overthinking the breech plug. There is really no reason to pull it out after every shot. Running the patches cleans the bore out and that is what affects your accuracy. As long as the breech plug doesn't clog and your primer ignites the powder, you are OK back there. At most, I would pull it maybe every 10 shots, but I don't even do it that often. I really don't think you are affecting your accuracy at all by doing that. In my mind you are just wasting time. At least I have never seen a need to do it. Not pulling the breech plug has never seemed to affect my accuracy.