Without a lift, what tires (size roughly 31x10.50R16LT) would you recommend for on/off road all terrain (DC, Goodrich, Fierce, etc.)?

I don't know if they would be called 31s in measurement but Rough rider Radial M + S are great tires from my experience and I haven't heard any bad about Dunlop Rover tires with a beefy tread, but yes BF Goodrich makes good all terrain tires as well.

Tire size depends on well clearance u can always go wider but taller is another story my s10 is marked for 205/70r14's but i have 215/75r15's on it. Looking at the picture i would say taller is not an option for u but wider is, personally i like the goodyear's they have never failed me they have a good mud tire just in general look for a tire with deep channels the deeper the better just remember the deeper channel u go the worse ur gas milage on smooth terrain such as road or highway travel the deeper channels also increase noise and vibration on said smooth surfaces. Just do some research check them out online or go to a tire shop and look at them even talk to a rep. Use a quarter to measure the depth of the channel its the easiest way i know to measure the depth. And u wanna look for a tire marked mt or rt. Hope this helps