Baking – Pizza baking order

If you have to trowel the sauce on the crust it is too thick. There is a "right" consistency that will prevent the pizza from being too watery but will still have enough moisture to allow the cheese and sauce to fuse somewhat.

On the other hand, you could just ignore that the cheese isn't fusing. I've eaten thousands of pizzas in Italy and most of them are more watery that what you'd get in the US and the cheese usually does not fuse with the sauce. Probably because it's only in the 800°F brick oven for about 3 minutes! Enough time to cook the thin crust, barely melt the cheese, and that's it.

If you really want that cheese to fuse, I would suggest a slightly more watery sauce, and more heat from the top. Turn on the broiler for the last minute or so until the cheese starts to bubble. This will also help evaporate some of the extra water in the sauce.

For a deep-dish pizza, around 425°F is right, and so is 20–30 minutes. That's starting with cold dough (need to keep the butter layers chilled, at least for a Chicago-style pizza).

Cooking in an aluminum 3" deep cake pan is fine. I suppose cast iron should work too (though it'll heat slower, so might take longer). As has been pointed out in comments, the cast iron much greater heat capacity may be part of the problem; I'd guess preheating it would help. (You can just put the cast iron skillet in the oven as the oven heats, though depending on how long you let the oven heat, you may want it in for only part of the time—no idea what the optimal temperature for it is). Make sure to have plenty of oil under the dough, and also cook on a lower oven rack.

I'd guess that you're using too-watery toppings. The tomato sauce should be pretty thick, much thicker than you'd ever use on a thin-crust pizza. Vegetables may need sweating to get some moisture out. If nothing else works, partially cook the crust (say, ten minutes or so) and then add the sauce and toppings.

I can vouch by Cook's Illustrated's Chicago-Style Deep Dish Pizza recipe. Normally they have a paywall, but currently that recipe isn't behind it, so grab it quick.