A 8" wide wheels with a 12.5" wide tire is a good combo for keeping your beads seated. The sidewalls being tipped inward toward the wheel make them want to stay in place. I have 315/75R16s (pretty much a 35x12.50) and have 8" wide steel wheels. I have been airing down a LOT for the last few years. It is amazing how much better the ride is and how much traction improves.
For the air pressure when running on the highway or pavement I took my kid's sidewalk chalk and drew lines on my tread sidewall to sidewall. Drive forward 70' or so and check out the chalk mark on your tread. The chalk wears off where there is the most tread contact. I found that at 40 psi I was only riding on the very center of the tread. My street pressure is around 27 lbs. My truck weighs 5,500 according to the local dump.
As far as pressure offroad- on some of the rough dirt mountain roads and I'm driving at speeds of maybe around 30-35 mph I'll typically drop to 15 lbs. If the road is more of a trail, then 10-12 psi.
Going slow thru very deep snow- I've dropped down to 5 psi. I was VERY hesitant to do this- but was talked into it by the rest of the group. Works amazing. I would drive at 10 psi then start getting stuck. No way could I go on. Hopped out and dropped to 7 psi. Was able to drive right out. Got stuck again- then 5 psi. Was able to make it to the top of the mountain then. On all of our outings no one has lost a bead- yet. It had happened a few times before I joined them. They said it isn't much fun, but not too bad to put a tire back on.
On my previous mud terrains I didn't air down. My tires were getting pretty chunked on rocks, sticks, gravel roads. With my new mud terrains I don't have any cuts/chunks. Airing down provides a lot more protection for your tires. Instead of a sharp rock or stick stabbing into your tire the tire is now soft and flexible and more likely to "ooze" around the object and keep going without damage.
If your tires are aired down pretty low they will want to heat up a lot if you drive too fast. An overheated tire will cause tire damage and can make the tire fail. Keep that in mind. More than likely if you lose a bead it will be on the front tire and the outer bead. Turning fast and sharp puts the most pressure on this area. So try not to turn fast and sharp.
You will need a good air compressor to air back up once your done. I bought a MV-50 as it was pretty much a cheap entry level compressor that can do the job. And it does work. I am looking into a better one as it is a bit slow (maybe a VIAIR 440P).
Different tires flex different amounts. The BFG KM3s I have are very flexy. Some run off road bias ply tires and they seem to not like airing down as much.
77K20 was spot on with the airing down and narrow wheels with wide tires definitely help to keep the tire on the rim without beadlocks. My old truck had a Detroit locker in the rear and ARB up front. The small ARB compressor works good to air up tires if you have a storage tank.
I 100% agree that a narrower rim will keep a bead better than a wider rim.
BUT, I've always wondered about tire flex based on installed width. I'm no tire engineer, but I would almost bet that any given tire will have more flex on a properly sized rim for any given psi, compared to having a tire mounted on a rim that is narrower than said tire.
So let's say a 12.5 width tire on a normal 8" or 10" rim. Let's just suppose that the lowest safe pressure is 15 psi.
Then let's say that same 12.5 tire on a stock width 6.5" rim. Let's just suppose that the lowest safe pressure is 10 psi.
My hunch is that the tire may flex the same or better on the wider rim, despite having 5 more psi, compared to the same tire on the narrow rim. The reason being, the narrow rim is tugging the sidewalls in opposite directions, where as the wider rim tends to keep the sidewalls on the same plane and more free to move about.
Again, just a hunch. I'd be interested to find out if there's been any comparisons like that, in an offroad application.