I also leave the exhaust brake engaged most of the time. On some models (especially VGT type turbochargers) you need to engage the engine brake frequently or the soot will bind it up over time (100,000miles). This was a common problem with the non-DEF Cummins. Wear and tear. Yes, the transmission is stressed more. On long downhills and heavily loaded I watch my trans temp (the torque converter generates a lot of heat). Only saw the temps rise once (10 miles of 6%).
Maintenance is more expensive with a diesel. Just look at how much oil they hold.
The cooldown part is important. The turbos get extremely hot when under load so you want to let it cool down a bit before shutting down (which stops oil flow to the turbo bearings). Turbos spin a very high RPM (up to 100,000 rpm) and need good oil flow. The owners manual will have a chart that shows cool down time for various conditions. When towing I usually do not shut the engine off unless I am stopping for more than 15 min. Yes, when fueling I leave the engine running. Stopping at a rest area we take turns and the engine runs.
I second the block heater for cold weather. Most wear takes place when starting (cold oil). Keeping it warmed is good insurance.
Tow/haul mode depends on the trailer/load. Our 7000lb trailer I never used it and rarely had to manually select gears. The 11,000lb trailer I engaged it all the time and on some hills would manually select the gear.
Have fun with the rig. Towing with the diesel is almost relaxing.