Congratulations on a job well done. I especially liked the looks of pizza #2.
The problem you experienced with the thin dough spots is a common one. When it does occur in the professional ranks, the most common expanation is inexperience--where a novitiate dough handler concentrates too much on the center of the dough rather than the edges. Unfortunately, when a thin spot develops, there is little that you can do about it. You can't scrunch up the dough around the thin spot and you can't easily use a "patch". I, too, have experienced thin spots and what I have found useful is not to press the dough outwardly from the center when I start to expand the dough ball outwardly with my fingers. I leave the center pretty much alone and press the dough outwardly away from the center. And when I stretch the dough, I stay around the edges as much as I can. Doing this is not always a walk in the park. If a dough is highly extensible, it can get away from you before you can prevent a thin spot from developing. It really takes a lot of experience to master dough spinning.
You might be interested in knowing that Tom Lehmann says that there is a functional purpose to tossing and spinning a dough. He says that during tossing, the air dries the dough as it swirls across the dough, forming a thin skin on the surface of the dough. He adds that the drying action helps achieve a crispier crust when the dough is baked. (So, dough tossing goes beyond mere showmanship.)
I have found tossing a dough to be most useful for a NY style and similar doughs with a lot of gluten. A dough made with only 00 flour cannot really be tossed. It can be stretched and shaped a bit using the fingers and knuckles, but not tossed into the air. A 00 dough is best shaped on a flat work surface by pressing the dough outwardly from the center using the fingers. Your combination of 00 and high-gluten flours puts you somewhere between the two extremes. I found that I was able to shape and stretch the Di Fara dough clone fairly easily, but I was very careful when I did it.
Another approach for shaping a dough is to do as you say you plan to do, that is, shape the dough without using your fingers and knuckles. I showed an approach like that to my daughter-in-law, who was not particularly thrilled about tossing dough in the air. The approach I showed her was to start pushing the dough outwardly from the center to the edge using the fingers and, when the dough was pushed and pressed out to near the desired size, to hold the dough down with the left hand about a few inches from the perimeter of the dough round, and then tug and pull the dough outwardly with the right hand to the final desired size. I saw her use this technique recently and it worked like a charm, with no noticeable negative effects on the finished product. Of course, this procedure should be done on a flat surface, such as a countertop or wood peel, not a screen. Doing this on a screen could force the dough into the small openings in the screen and make a real mess of things.
As for the matter of crust thickness based on using the 00 flour, I have heard and read of many descriptions of the thickness of such crusts--from "thick" to "thin", to crackery, etc. But, since the terms "thick" and "thin" are relative, I have come to rely on the recipes I use for such doughs and the thickness that comes from making the doughs in the recommended sizes (diameter). I have found the 00 crusts to be thicker than the NY styles, and softer and chewier. It's purely a matter of personal taste whether you prefer those crust characteristics over a NY style crust. I personally put the two styles in separate and distinct categories and don't try to compare them in terms of overall preference. I like both equally well.
I was amused to see the looks of your pizza #3 using the KA 00 flour. It looked like the ones I made using that flour--and found wanting. Some time you will have to try other 00 flours and draw your own conclusions.
I assume you noted your formulation for the Di Fara dough clone. Can you post the formulation some time? I'd like to give your version a try.