>>2796584>hatchetJust get a cheap hardware store hatchet.The brand-name stuff is no better nowadays (or even worse - fiskars and estwing suck), and if you knew what to look for in an old hatchet, you wouldn't be asking here.
>SawSilky. I'd get the F180 (fibreglass handle) since it's a lot cheaper than the full-steel models, unless you're into using minimal gear like I am. In that case, get the small pocketboy. I've got both in use, as well as a katanaboy, and those saws are way better than all others I've tried.
>Camp stove/cooking systemDepends on your location and laws. If you're in the far north, a folding gridle and an open fire will work just fine, as would a gasoline stove, while a gas or spirits stove wouldn't even light. On the other hand, in the subtropics, an open fire means risking a wildfire, and a gasoline stove might evaporate enoug fuel through any leaks to become dangerous if not perfectly maintained.
If you're above the freezing point, I'd stick with a simple spirits stove (Esbit or one of the chinese ripoffs, as Trangia is less efficient and more expensive. Look for one that has two different hole sizes). Then get a folding windshield / pot stand combination. Most of those double as solid fuel stoves for sub-zero temps.
If you're below the freezing point often, get a gasoline stove. Those are expensive, but the only thing that'll light reliably in those conditions. Spirits will freeze just a little below 0C, petroleum works in principle, but needs spirits or gasoline to preheat, and if you've ever tried to build a fire with frozen wood, you know that it's a pain.