To the first question, the only way to truly know the dry weight of his rear axle is to weigh it. By comparison, my 2010 F250 with a V-10, crew cab, short bed tipped the scales at 3140lbs.
To the second, the tire pressure on the door sticker reflects the OEM tires' recommended pressure for the maximum axle loads. So, for example, my door sticker lists the max pressure as being 65psi because, at its 6600lb rear Gross Axle Weight Rating, that's the air pressure Bridgestone recommends running at to support 3300lbs per tire. The tire is rated for something like 5600lbs @ 85psi, but there is no reason to run them at the max pressure on the sidewall if they're never seeing that load.
That being said, I run them inflated to the pressure for the load. So, when it's empty, the recommended pressure is something like 55psi. Loaded for a weekend of camping, trailer in-tow, they run at 65 if my memory serves because, loaded-up, they are seeing 4280lbs, and that's the recommended inflation pressure for the tires under that load.
So, all that being said, adjust the tires for the load they're seeing, plain and simple.