I'd recommend Cooper AT3 or Goodyear Duratrac in a medium width tire. Narrow is best for Winter roads, but wide is best for sand. I personally have found medium width tires best for mud.
You need a medium width tire (relative to your vehicle weight) to deal with the variety of terrains. For an F150, I think you need a tire 11 to 11.5" wide.
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Also look at Hercules AT2 and Hankook ATM.
Any of the tires I mentioned would be good for you, IMO, but I'm not sure about the Hankook ATM for mud. No idea how it does in/on mud. It is known to be great for Winter roads and sandy/rocky conditions though. I speculate the Hercules AT2 would be great, but I don't know for sure.
I own the Cooper AT3 and I know it's great on everything, except on ice it's rated OK. I've never had it on ice. It's sure good on highway dry or wet, and on rocks, mud, and trails. I haven't tried it on sand, but its AT tread should be fine on sand.
BFG AT would also be good for you, but some of those other tires are (IMO) even better and cost less. I owned BFG ATs before, 2 set of them. They were great at almost everything, but only marginally OK in mud.
General Grabber AT2 would be good for you too, IMO, except that it has a hard/stiff ride (if you care about that). I previously owned this tire for a while, but I hated the stiff ride and traded them in for some Cooper AT3.
For softer, pleasant ride quality on bumps, Cooper AT3, Hercules AT2 (cousin of the Cooper), and BFG AT are the best, IMO. The 3 things I didn't like about the BFG AT was high price, hydroplanes easy on wet Interstate at 65+ mph, and not to good in mud.
So that leaves the Cooper AT3 and Hercules AT2 as the best tires that also ride really nice. While I am speculating on the Herc AT2 since I've never owned it, I do own the Cooper AT3 and they are the same tire, except for a few differences in tread and sipes. The Cooper AT3 is quite good in mud and snow and on highway too, but has only 2 sipes per treadblock which might explain why it's only rated OK on ice. The Herc AT2 is same tire carcass with similar treads, but has 3 sipes per tread block, which is why I think the Herc AT2 would be better on ice. No idea how the Herc AT2 would do on mud, but I'm sure at least as well or better than most of the other ATs mentioned.
The Duratrac is (IMO) more aggressive than an AT. I think it's between AT and MT. It might be just what you need for all the off road terrains you mentioned and for Winter roads too, but might be less pleasant at high speeds on highway than an AT. A Cooper AT3 probably isn't as good for mud, but it's pretty darn good in mud, and excellent on highway, IME.
I think Cooper AT3, Hercules AT2, Hankook ATM, or Duratrac are the best choices for your needs. The Hankook ATM being the least aggressive tread. The Duratrac being the most aggressive tread. For what you describe, I'd choose the Cooper AT3, Hercules AT2, or Hankook ATM.
Another consideration is the tire has to be available in a size and load rating you want. I recommend load D because plenty tough enough for 1/2 ton F150, yet still has a comfortable enough ride and decent gas mileage (compared to E). Only get E if you want max toughness (more than you need IMO) and are willing to sacrifice ride quality and gas mileage.
Is the tire of your choice available in load D in a size you want? That's an important consideration. What size tire do you want?
P.S. - I would NOT recommend Cooper ST Maxx for Winter conditions. It doesn't have enough sipes and it's rubber is very hard, both of which would make it very slippery on ice and packed snow. i.e. - snow that someone else has already driven on. I owned the Cooper ST (prior version before the ST Maxx) and it had few sipes and hard rubber. It was extremely slick on Winter roads. Then I paid tire store to add more sipes. After that it was OK in Winter, but never good. A tire with hard rubber (most commercial tires have hard rubber) is always slick in Winter, even with lots of sipes, but more sipes does help. The ST Maxx lacks enough sipes, and has hard rubber. It's a no go for Winter roads IMO.
Warning: I often edit my posts a few times to get them complete or correct errors.