Duramax 2006 is called the LLY with the 6 speed Allison, no DPF. DEF wasn't intro'ed until 2012, mandated by the EPA. Body style is called the GMT 800, or "Classic". There were some late releases in 06 with the LBZ designation.
Duramax 2007 "Classic" is the LBZ. Same motor/tranny as the 2006 LLY, but with a hotter tune, larger radiator. No DPF. GMT 800 Classic chassis. There are also 2007.5's, with the newer GMT 900 body, stock with DPF. Same tranny as the 07 Classic.
All the issues, and I mean all, have been worked out with these trucks, and all the weak points hardened. There is virtually nothing you will run into that doesn't have a better than stock fix, motor, drivetrain, chassis. From the front ends, CP4 injector pump, PTO cover, transfer case pump rub, all solved and won't kill you to fix if necessary. But unless you have very high confidence that the truck hadn't been run with a hot tune, anything over 500hp at the crank, be prepared for a tranny build/update. The Allison can take 500hp stock, but needs to be hardened above 500. Stock with good body these trucks are gold. GM haters will argue, but close to stock these truck were the best motor/drivetrain/ride/mileage/durability on the road. Truck of choice for hotshots sums it up.
Frankly, I wouldn't hesitate buying a GMT 900 07.5-11 if it was DPF deleted but not hot tuned, or I had a good delete kit waiting for me. These deleted trucks are solid, just as reliable as the LLY/LBZ. GM haters gonna hate, they all made some good trucks and problem trucks, all with weakpoints. I'm a fan of the Cummins motor, and the Super Duty 6.7s are solid, just pricey. You know the diesel vs. gas game over the long haul already. Good luck.