I've always been a big outdoors guy and have done a lot of hiking. Usually I'll end up carrying a smaller folder and mid sized fixed blade combo. Throughout the years the knives have changed a lot, and continue to change all the time just because I like to have excuses to carry my knives. I carried a ESEE4 with a SAK ranger banded to the sheath for quite a while. The passed couple years my main 'beefy' outdoors fixed blade has been a Becker Kephart with the micarta slabs, I really like it. That's my camping knife, and if I were to need to possibly do some light clearing of the stray blackberry vine here and there and what not, that'd be good for hiking too. It'd probably be a good one for your longer trips like you're talking about. However lately I've been trying to go more lightweight on my hikes, and usually carry a light(ish) folder, and a lighter, smaller fixed blade. Here's a few examples of what I've carried lately.
I love these two little compact sharpening rods. One is diamond (EZE-Lap I believe, screws into the brass handle) and one is ceramic (Worksharp), a perfect combo.
This is a super handy, lightweight fixed blade I got eons ago from Landi.
On this one I switched it up to a tough folder and a lightweight chisel ground fixed, cuz why not. I've never messed with the foldup wood burning stoves. They're cool as hell, but I generally like to just get r done when I'm making something or want a hot drink. I've been using this MSR isobutane stove for 20 years or so and it's been awesome.
I've used this Allen Blade MEUK a lot as my hiking fixed blade too. He makes a hell of a knife, but he's disappeared as a maker I think.
And a shot of the ol' Becker Kephart when I got it.
But yeah I agree, karate would probably be better.