Wow, there's a lot here.
Okay, I don't have a lot of guns, but about a dozen or so. Here's some things:
1: squirrels are tougher to kill that bunnies.
2: In .177 I'd be REALLY iffy on taking even rats at 10 yards with anything under 600fps with a 8 grain pellet. a *tuned* quest 1000, hammerli storm, or gamo equivalent will do you, my quest gets 830 fps with 10 grain crosman heavy pellets. Storm is right at 815 with the same, but shoots a tad smoother. That's fine for closer range hunting. My customized 1377 american classic pistol hits about 680fps and that's FINE for close range hunting, if you get head shots.
3: I'd suggest .22 for a hunter- really strongly suggest. The delivered energy is much, much, greater. I've had .177 pellets pierce animals and just go right through without killing. (The exception is Crow Magnum pellets, which cost more to shoot than a .22LR) - get a .22 I won't argue a lot with a .20, but it's not as common and the benjis aren't as easy to fix.
4: Co2 is fine for gathering food but not for EOTWAWKI. In general to get CO2 to perform you need to customize a gun a bit. A stock crosman 2260 is a decent hunter but you can gain a huge boost in performance with a few upgrades from an airgunsmith. The 2250 is my favorite among easy .22 CO2 guns, once you put a steel breech on and a BOSS valve. Also, with CO2, larger caliber == more efficiency == more power.
5: I tend to avoid spring guns, and the newer gas ram spring guns- a lot of people swear by them BUT- with springers (not so bad with gas rams) you have to decock at some point. If you go out hunting and leave the gun cocked for too long, you lose spring power. And it *does* happen. Even with the $600 german guns. Springers are also very hold sensitive due to the recoil path- and you CANNOT get the accuracy you deserve with regular rifle holds. No way, No how. Springers are also much more difficult to maintain on your own. They are AMAZING performers if you do everything right and a .25 gas ram Falcon would make me happy up to turkey sized game.
Pumpers- the multi pump pneumatic is IMO the lightweight, bushcrafty or backpacker's survival airgun of choice. They *tend* to be smaller and lighter than full power springers, and don't rely on CO2. A complete rebuild kit for a crosman pump gun weighs under a pound and will take care of anythign short of massive breakage. It's also bone simple to rebuild one. The benjis (sheridans, Benjamins) are generally considered the touchstone for pumpers- and my benji 392 stock rifle can hit 615fps real world with 14.3 grain hollowpoints, and is still over the critical 500fps with heavy crow magnum ammo. My Steroid tuned benjamin can rocket over 800fps, well into serious 60 yard hunting territory. But they are big and heavy compared to the crosman carbine possibilities. I have several pump guns based on the 1377 pistol. With a folding stock, a .22 caliber conversion, and some valve work, you can get a good solid 600fps on 12-14 EASY pumps (far easier than the benjamins and sheridans) from a 12 inch barreled pistol carbine. at 1/3 the weight.
Finally- don't discount the PCP (air tank) guns. These days, with a specially made hand pump, you can get a benjamin Discovery kit, complete set up with pump and rifle, for $400. You get more power, more shots, and more comfort than any pumper, for about the price of a top of the line custom pump airgun.
Remember that once you go above .22, larger calibers == more money. Sometimes a LOT more money. if you really want a large bore, look at some of the custom conversions available on basic crosman frames- I've fired .25 caliber pump pistols that deliver a ton of energy, and a CO2 carbine in .32 that shot round ball and would just SMASH any small game at reasonable ranges.
Resources for airgunsmiths are available - just google and you'll find eveyrthing you need, from target to .50 caliber big game airguns.
If you go looking through gunsmiths you will sometimes see used rifles. the crosman 101, 120, 140, are all excellent day to day game getters.
Here's Koyote Girl's 1377 with a stock and a set of custom grips. Scoped in this photo, with a 10 inch .177 barrel. It got abotu 560fps max at that time:
After I did valve work, put on a 14 inch .22 caliber barrel, and showed her the peep sight (now it gets ~600fps in .22)
an older 120 I'm working on. Shoots good, still needs a bit more rebluing and I could stand to reseal the breech a bit better:
and my crosman 101. Made in 1926, shoots .22 pellets around 580fps on 8-9 pumps. DEAD accurate with the tombstone peep sight. My favorite. handles, shoots, and acts like a .22 rimfire.