I own a 2001 Silverado w/same configuration. It has a 7800# tow rating. I greatly improved performance, milage and low end torque by installing a K&N air inlet assy and a Raven cat-back exhaust. the difference is amazing.
I have a GMC 2500HD with the 6L engine, and it is marginal for towing about 6000 pounds of fifth wheel. It's a rock solid platform for towing and braking at all speeds, and rides like a car with no trailer, but it has trouble getting up hills at highway speeds. No heat problems though; neither the transmission temperature or the engine temp twitched no matter what. On my recent trip, I spent most of my time from OK City to Albuquerque in lower gears. I haven't done the math to be sure, but it appears that gas milage dropped to near the 6 MPG point for this part of the trip (9 MPG with trailer and minimal hills, 13 MPG unloaded around town).
I got the trailer to tow with my 89 Silverado with the 350 engine, but just towing it home was more excitement than I needed; it was right at its GVW/CGVW limits, plus had over 200K miles on it. The new truck was a good idea, I just wish I had gotten the bigger engine.