Forget about a folding bike, they simply aren't built for that terrain unless you're talking about something like a Montague Paratrooper. Many folding bikes have 16" or 20" wheels which are going to dig into the mud and get bogged down. The only real advantage a folding bike has for commuting is you can fold the bike up at work and store it under your desk, but if it gets real muddy you won't be doing that anyhow.

I wouldn't recommend a mountain bike because they are made for aggressive trail riding, like riding down a mountain or on a mountain bike track. Check out Youtube for mountain bike riding vids to see what those bikes are designed for. The bike's frame geometry and the rider's body position isn't the best for long rides on the road. Sure, lots of people commute on mountain bikes, but you don't necessarily have to if you're buying a new bike.

If you look at Surly bikes, some of the frames like the Long Haul Trucker will accept 26" wheels with fatter tires than normal touring bikes. I wouldn't want to ride on 700cc tires in the last two sets of terrain you had pictures of, no matter what type of bike they are on. It's a trade-off: wider tires are better for off-road and less efficient when on paved roads. You'll have to decide where the dividing line is for you.

A cyclocross bike would also fit your needs because they are meant for both road and trail riding, albeit not the same type of trail riding as mountain bikes.

I've got a Salsa Mukluk 2, and although it isn't the type of bike you need, I can say that I am impressed with Salsa in general - well designed, good component quality, etc. You should look into the Salsa bikes the others have recommended.

You'll also want to get a geared bike and not a single speed. It sounds like you want to do some exploring, and a single speed bike isn't going to cover the same amount of ground or climb the same hills as easily as a geared bike will.

The last thing I'll say is that as someone who gets all of his groceries on a bike, you'll want to get a rack if you plan on doing the same. The backpack isn't a good option when you're carrying a gallon of milk and/or a jug of orange juice along with other items, the weight will become uncomfortable after a short while.