I agree with the concept and price point. A lot of the cheap guns I have are purposed as "stash guns" but I won't go into further details.
.22LR:
Mossberg Plinkster. This is a sleeper budget .22 rifle. Really stuck with 10 rd mags only, but it's $130 brand new, less if on sale.
Marlin Model 60. These can be found used cheap. You can use a spee-d-loader to quickly load the tube magazine.
Henry .22. A bit more money, but the quality is worth it and with this, you can use the very quiet .22 LR ammo and it won't affect cycling.
12 ga:
Pretty much any single shot shotgun from Savage, Hatfield, or Midland is gonna run 100-150. I like the Midland because they're coming out with barrels later this year for .22, .357, .223, etc. I'd imagine the barrels will run for around $100. If you're okay with a single shot .22, this is an option you should keep in mind.
For pumps, there's only one option and it's the Mossberg Maverick 88 field and home combo barrel.
Big game hunting rifle/scope:
Given your location, you're potentially dealing with VERY big game. I've recently been turned on to the .35 Whelen cartridge and I think that would suit a lot of your needs and it's available in cheap rifles like the H&R Handi or CVA's. I can't say the same about .300 Win Mag or .45-70.
The other option here is a Mosin-Nagant. Ever since I was a kid I was told the 7.62x54 can kill anything in North America. Very cheap if you can catch a deal on Mosins these days, but that market is drying up.
I have no idea what scope to recommend. If you want something to stand up to the recoil of the big game rifles, you'll have to spend money. My suggestion is for the big game, use a shotgun and slugs and for the not big game, a .308 Savage Axis or Rossi Wizard or H&R Handi or Mosin Nagant.
A real dark horse here is a .54 caliber muzzleloader and some Hornady Great Plains bullets; it will put a lot of big game down. Round balls will be great on deer too and it's all entirely too cheap.
Handguns:
Hi Point in .45 ACP comes to mind. That is a terrific stash gun given your location. Don't discount .45 FMJ as a defensive load for bears, it penetrates, and for cabin defense, it'll work great as gun under the pillow.
If you want to carry something that's not a brick while you chop firewood, any of the budget 9mm's will do. I think Ruger 9E's are a steal and if you want something smaller, Ruger EC9s or LC9s is it. If you want even smaller than that, the pocket .380's like the LCP are going for under $200 now, but if you're comfortable with .25 ACP's, Raven MP-25's can be had for $100.
For revolvers, I see a lot of used Charter Arms go for low prices and the older ones are great. I got a .357 for $250 last year and it's awesome. The .38 snubs are $200 wonders, the .44's are about $300 but I wouldn't bother due to ammo price and because of something I'll touch on later.
Heritage .22 LR/.22 Mag would be your best choice for a cheap .22 revolver. They have 9 shot models and various barrel lengths. My suggestion is to buy one with adjustable sights because the fixed sights suck. It's worth the extra $100 to be able to hit what you aim at.
What you shouldn't write off is black powder cap and ball revolvers like the 1858 Remington. If you can't store a lot of ammo and you're not going to reload, you can shoot thousands of times with a cap and ball revolvers with a few pounds of powder, balls, and caps. When Cabela's has them on sale, you can grab an 8 inch 1858 for $200. Spare parts kits cost maybe $40 and if you ever have something fail inside it, you can fix it cheap. Spare cylinders cost $50, you can load and cap those spares and carry them like speedloaders, and the 1858 is accurate. You can also get conversion cylinders to shoot .45 Colt and .45 ACP, but lead bullets only.
Do the materials for cap and ball revolvers cost money? Yes, but it gives you an option to always be able to shoot back if you need to shoot back at something and a .44 cap and ball is nothing to sneeze at, they're just as powerful as standard pressure 9mm or .45 ACP is today.