I have a 2004 Suburban 2500 that I've been playing around with using EFILive, especially around the transmission tuning. The stock transmission tuning, at least for my 8.1, is very aggressive (much more so than the 6.0 tune), and changed year to year across all of the (few) stock calibrations for Burb 2500s I've been able to find.
At least with the 8.1, the converter has a fairly high stall speed. There is some economy to be gained by locking up earlier at light load and lower speeds - I picked up over 1 mpg around town which doesn't sound like much, but it's a 10% improvement when you're barely gettign 10 mpg city to start with.
Unlike the 4L60, the 4L80 doesn't seem to have any problem running unlocked in 4th (4l60 cuts cooler flow in OD and gets hot if the converter is unlocked)
The 8.1 tune (and if memory serves, the 6.0 tune as well) locks the torque converter in 2nd gear over 93% throttle, and will shift with the converter locked into 3rd and 4th. The converter will also lock in 3rd if you're on the throttle pretty hard, and will stay locked when it eventually goes into 4th. This is usually a pretty abrupt shift.
I would not lock the converter in 1st. I don't think it's even an option in the software.
Locking the converter is fine when towing; it really improves efficiency and reduces trans temps. You just don't want it locking at low engine speeds and lugging the engine at high load, and I personally really don't like the locked shift from 3rd to 4th. It's pretty aggressive and pretty abusive.
There is a switch in the software to allow lockup when shifting. It doesn't do what the internet seems to think based on my testing - the PCM already allows the converter to stay locked regardless. That switch allows the converter to lock at the same time as the upshift if it wasn't locked already and the TCC table for the next gear is commanding lockup. I would recommend not turning this on - let it upshift and then lock up.
The internet also has a lot to say about torque management. Don't get rid of it, and if your maps are allowing shifts with the converter locked, I'd really suggest making the torque reduction even more aggressive than stock at high load. Combine that with slightly higher line pressures and you will get quick, positive shifts with less heat generation.
Something to think about when tuning the converter that took me way too long to wrap my head around: there is no "default TCC off" when shifting into the next gear. The TCC will remain at whatever state it was previously in unless you are below the TCC release speed in the gear you just shifted into. So if you want to go from 3rd locked to 4th unlocked, you will need the TCC release speed table to be above the 3-4 upshift table which is, generally speaking, not practical because you'll be running unlocked far to often to be efficient. I think most people just move 3rd gear lockup out of the way, but I don't like that option much either because, when towing, it's worth about 1-2 mpg if you are pulling a hill in 3rd locked vs unlocked and generates a lot less heat when locked. I just let it shift when locked with very aggressive torque reduction.