I think I included everything important???
One of the most important numbers you left out. The trucks FAWR/RAWR as they carry the load.
However looking at Fleet Ford 2015 F150 weights specs show a 7000 GVWR F150 has a small 3800 rawr..... (rear axle weight rating) carries all the weight in the bed and is good for around 1400-1500 lbs in the bed.
The next step up is the F150 with a 7050 GVWR and 4050 RAWR good for around 1600-1800 lbs payload in the bed.
Another step up is a 7500 GVWR and 4550 RAWR good for approx 2100-2200 lbs in the bed.
At the top of the weight chain is the F150HD with a 7850 GVWR and 4800 RAWR with around 2400-2500 lb payload in the bed.
All weights are a estimate for math examples. A crew cab F150 is hevy and has less payload than a std cab or super cab.
From what I read all the different GVWR/GAWR packages can be had with a max tow package. Maybe some Ford folks have the skinny just what the F150 max tow includes.