I had friends who used to full time with a E-350 van that was new in 1979 and towed their 30' Airstream. Later they upgraded to a 34' Airstream, and I towed that to a storage facility when they sold the Airstream. It towed really nice, and had plenty of power. The 460" engine was only rated at perhaps 200 HP back in 1979.
You will have plenty of power. If you really feel like it, and floor it, you can make it up the mountain grades at 60, but why waste the gas? 55 or 45 is plenty of speed for going up the 6% mountain grades. Doing 70 down them is kinda crazy, so why waste more fuel to go 70 up them?
If you find that you need to frequently hit the brakes going down a hill, make sure that you lock out overdrive to assist in slowing the truck/trailer combo.
AS for capacity, any travel trailer with up to about 1,400 pounds of hitch weight is easy enough for any 3/4 ton truck to tow. That limits you to about 12,000 GVWR on the trailer. Yet if you start looking at fifth wheels, they put about 20% of the GVWR on the pin, so only about 8,000 pound fifth wheel will normally have a 1,500 - 1,600 pin weight, more than your truck can 'comfortably' handle without exceeding your GVWR of the truck.
My friend's 34' Airstream was about 10,000 - 11,000 pounds ready to live in travel, and probably had about 1,200 - 1,400 pounds of hitch weight. It was 3 axles, and towed by a now vintage van. They did not have fuel injection, only had a 3 speed automatic transmission. Your set up is much more powerful than theirs, and their gross combined weight would be around 18,000 pounds! Your truck is probably around 7,500 pounds and trailer 7,500, so you will be close to 15,000 Max gross combined weight. You will be fine.
As for backing up a larger and longer RV, that is another problem, but should be easy enough once you get used to it.
Pulling into a gas station is easy too. I used to pull my 30' motorhome into small gas stations and fuel up, pulling my 15' long car. I look for open areas around the pumps, and plan on pulling through, not needing to back out ever.
Good luck,
Fred.