Too much goofy speculation. From what you said, everything is 100% fine.
Couple inches of sag is to be expected with a moderate load in the bed. F250s have a rather soft initial spring rate and stiffen up after that. You can add whatever you want to the back to stiffen it up more if that's your desire, won't hurt anything.
No your AT tires are not making it drive bad (save for maybe a bit of tread squirm when brand new). It will drive slightly different, lol, with a 5th wheel trailer hooked up. Some folks don't understand that.
You're good, go camping, have fun!