Loo and BTB are the experts on the Home Run Inn style pizza but I believe the answers to most of your questions are found in the opening post and in Reply 2 in this thread. As I interpret Loo's instructions, the sauce goes down first, followed by the toppings, and then the cheese. Also, if docking were an integral part of making the crust, I believe that Loo would have mentioned it. Since he didn't, I would not dock the skin.
It sounds like the most useful pan you have for the HRI clone pizza is the 10" cutter pan. Based on the information provided by Loo in the opening post, the amount of dough you would need for that size pan is 3.14159 x 5 x 5 x 0.111 = 8.72 ounces. So, if you made the full amount of dough according to Loo's dough formulation, which I calculate to be 17.10 ounces, you should be able to make two pizzas slightly less than 10" in diameter. You might be able to use the other pans, in which case you will have to calculate how much dough to use for those pans, using the simple math expression given in the last sentence but using the corresponding radii of the other pans. I suspect that it is possible to use a screen, either alone or in conjunction with a preheated pizza stone, but you may have to dress the pizza quickly if you use a screen because the combination of 42% hydration and 24% corn oil may translate into a dough that is quite wet and likely to stick to the screen. I believe that it is because of the wetness of the HRI clone dough that Loo used a disk or pan instead of a peel and pizza stone.