Adam, CB has listed the platter thickness in the BS page - IIRC 1/4 inch. For your use, I would think you could get by with something in the order of 1/8 inch, there is not a lot of load that needs to be supported, and it will be easier to weld the drive rod to it. Think about how thin your pizza pans are, and if you stacked one on a top of quarters ( simulating the drive rod ) and loaded a pizza, the pan wouldn't bend or droop appreciably. If you read the first pages of the BS thread, one problem was one or two users turned it on to heat up, and came back a short time later and they had melted - burned the platter because they had not turned on the motor for the rotiss. At first I found it hard to believe, but then if you think about it, it is just a big propane torch in a small space - it will definitely take most metals and get them more than red hot pretty quick. So my guess is they made it that thick to keep from that from being a major problem. Many of us do not use the metal platter to distribute the heat, since we put spacers so the heat is distributed by air, not contact with the metal platter.
In terms of height, pretty easy - cut up some cardboard boxes to different sizes and try loading and unloading pizza - and use a size you feel comfortable with. I use a super peel which makes it extremely easy even with very little clearance - and I had a prior electric oven that I modified and ended up with maybe 2 inches of space between the bottom stone and the top element. My current electric oven has about 3 1/4 inches ( though I don't use it since I have the BS ) Definitely the higher it is, the easier it feels to launch it, so I would err on going a little higher than you absolutely need to avoid any stress. Also, the higher the opening, the deeper you can see into the oven to see if there is any burning cheese.