Just to quickly touch on this, at least the bits I have experience with:
3) 31x10.5 will be about as big as you can go on stock suspension/wheels. (There are a couple exceptions to the stock wheel rule; but most of the stock rims will only safely accommodate a 10.5" width tire). You'll also have some rubbing on the control arms if you turn the wheel too far left/right - you can adjust the stops if you like, or just not turn the wheel too far.
5) No. You will lose some power - I'm running 31x10.5s on the same transmission/differentials and I simply don't use 5th gear anymore
70mph in 4th isn't a problem though.
6) To avoid rubbing, yes. You can get wheel spacers, wheels with a deeper offset, or you can pull the steering stops out, add a couple of washers for spacing, and screw the stops back in...(or, just don't turn the wheel that far...lol)
7) Yup. I picked up my 31x10.5 BFG MTs a month or so before installing my 2" lift. They fit fine, but the added room from the lift definitely paid off.
Essentially you can run 31x10.5's on an otherwise stock Jeep without any further mods being necessary. That said, a 2" coil spacer kit is dirt cheap, and if you're looking at replacing the suspension anyway ... just do it. Re-gearing, while also not strictly necessary, will be dictated by how you drive. In my case, I still had enough power in 4-Low for the trails I drive, and yet can maintain a good 18 to 20mpg on the highway, as well. If money is a concern, I'd put the 31s on and drive it for awhile like that, just to be sure I'm happy with the gearing. Your mileage may vary...(pun intended
I've been riding on 31s for 70k to 80k miles now, and haven't re-geared or addressed the wheel-rubbing issue, I just adjusted my driving habits instead.