Welcome to the board!
You should be able to fit a 31" tire on there with very little or no modification.
As long as the rear axle hasn't been midified, you'll most likely have a LSD in the 4x4 models. check for an orange dot on the rear axle housing. if you don't have an orange dot then it's an open differential.
the 4x4 doesn't work like a AWD, the front and rear are locked together 100%
There isn't a whole lot of aftermarket stuff for these trucks but you can always browse around the store here on Rugged rocks.
for the check engine light, run the codes on the computer and see what pops up.
Post any more questions you may have.