No doubt the 31X10.50R15 look small on the rig as that size fits easily with no lift at all. With the body lift you currently have running a 33X12.50R15 should work with no problems. Adding the suspension lift will make the 35s fit, however when turning near full lock they will rub on the frame (no big deal) reducing your turning radius & making a rude noise. You may be satisfied with the 35s without out a gear change but it will certainly make the rig a dog when you accelerate. Going to 37s you'll need gears & the rubbing on the frame will be worse & need a bit more lift.
As to the brand & style? of tires, I would suggest you give lots of thought to exactly what you want to do with the rig before you run out & buy something. If you are mostly concerned with how they look it's a bit easier & comes down to style & price. If you plan on actually using the rig put together a list of prioities you want to do before you start tire shopping that may end up costing you more when you buy the tires but greatly reduce the likelyhood of you having to live with something that doesn't work for you for your driving needs.
Everything a vehicle does is ultimately controled by the tires! GO, TURN, STOP, it's they only part of the vehicle that contacts the surface unless you've done something BAD.