I'm not sure about my rim width being 12, but I have wide 305-50R20s (Toyo AT2), basically 33"x20 tires, on my H3T and LOVE them. When I bought my truck, it had the 20s and I planned to swap them for 35s on an 16 in rims like on my 06 H3 when I got home. But after the 600 mile trip home on the interstate, I was totally sold on the uberwide tires with a narrower sidewall. It was a huge improvement from the high speed mushy wallowing of AT tires with big sidewalls. I'll take the rock steady stability at speeds 75 over the squishy big tire look that feels like "handling by Beautyrest". It may not the best tire for serious rock crawling, but for general driving where I do 99+% of the H3T's miles, the 305s are superb and the Toyo's stiff sidewalls still give me plenty of protection on the trails and FS roads here in N AZ.
Leveling gains you little; some room for over-sized tires, but robs torque and increases the likelihood for front axle problems. The suspension was designed for the nose down. CV failures seem a lot more common on "leveled" trucks.