How do I ensure my deep dish crust gets cooked
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.