Well I am no green egg pro, but I believe I clearly see a high temp at the bottom stone, and perhaps no top stone? Your pie looks like it should have been taken out earlier, but the top never browned and the cheese didn't really melt. Seemingly these top stones are key when using like ovens, and the gaskets blow and need to be replaced with something else, also common from what I've read. Here's my pal RS's thread on this, he mentions replacing the stock gasket on page 2, and you might find some good pointers overall.http://www.pizzamaking.com/forum/index.php/topic,8334.0.html
You're good having the mixer, it was not $300 poorly spent. You can knead more effectively, make higher hydration doughs, make bigger batches, all sorts of good things that I can't do easily if at all hand kneading. What it comes down to, aside from learning to use mixer right, is getting a feel for the dough. This dough looked to me like it needed more kneading and better forming in early stage pics. As it climbed the hook and all I'd guess it didn't knead well, or for long, in mixer. (The great Jeff Varasano makes a point to always start with wet ingredients and add flour slowly to avoid this climbing, for what it's worth.) Once you then re-balled it after warm rise, as it tore, that was it. Hard to get a good result after that. Trying to repair the tear is the best you can do. I'm surprised you got it to re-stretch right away, frankly, so hats off.
A tight ball, of properly kneaded dough, going into container is key.
Here's a couple useful vids on that, as I think some hand kneading at the end is always a good idea, even for mixer users. I stretch a bit before folding over as well, but here's the kneading method.
How to make a good ball, and I also slap the top down every few turns. Not hard, but enough to really make it tight.
Also when you take it out of fridge put ball on flour dusted board to rest. Use this same rolling method in video 2, no slaps on top though, to ensure roundness. Dust top and saran wrap it loosely, as Peter said. It's best.
Hope it helped, just keep at it, it all comes into place.