We have the same truck, a 2006 Tundra 4x4, and we have pulled Airstream about 14,000 miles so far with it. Our trailer is 800# heavier than yours, and we use an Equal-I-Zer hitch. It's a good tow vehicle but may have limitations in for some people, fine for us.
It is weak on power on steep grades, high altitudes, so shift down and reduce speed as needed to keep the engine in 3,000 rpm power range. On the down side of very steep grades I shift down and slow way down to ensure good braking reserve and that trailer does not want to go faster than truck and begin sway condition.
Never tow in overdrive, the cruise control will keep shifting the trans down and up. This will not affect the gas mileage when towing. Cruising speed is normally 60 mph, 55 in headwinds and 65 in tail winds. Expect 10-15 mpg.
The stock P rated passenger tires' sidewalls are too soft and cause the trailer to sway easily, can be very uncomfortable. I installed Yokohama 235/65 R17 XL (Extra Load) tires and the stability improved dramatically, and the slightly smaller diameter gave more power on grades. The ride when not towing is truck-like, rougher, but that's the price of greatly improved towing stability and worth it.
I installed a Prodigy brake controller with the Prodigy wiring pigtail, so it just plugs into the truck wiring, there is a connector under the left side of dashboard.
Works well as an Airstream tow vehicle partly because of the aerodynamics, independent suspension, and low profile of the Airstream. Would not be as good with a "box" trailer of the same weight.