DA lets you fire the gun without manually drawing the hammer back with your thumb via a long, heavier trigger pull. Most hunting situations are better handled in SA mode with either design.
The grip on a DA is higher, so recoil is more "straight back", while typical SA grips are lower, causing the revolver to "rock back" in your hand.j
I prefer SA revolvers for hunting, and DA revolvers for self-defense.
If you want to reload, then as you said, rimfires are not a good option for you.
In centerfire cartridges:
cartridges like the .25 ACP, .32 ACP, and .380 ACP are designed with small, compact semi-autos in mind. While the cartridges themselves may (or may not) offer what you might need, the firearms that chamber them aren't hunting handguns.
So I'd exclude them as options.
Someone above mentioned the .327 Federal Magnum. I think the .327 Federal is a great cartridge, but it's going to be too hard on small game that you don't want to "blow up". In a handgun chambered for the .327 Federal, you can also shoot .32 H&R mag, .32 S&W longs, and .32 S&W shorts with good results.
But, while these are all decent choices, firearms chambered for .327 Federal are, as of yet, fairly rare, and ammo can be limited in choices, and hard to find. I have a Ruger Single Six in .32 H&R mag (so it'll chamber any of the "lesser" cartridges, but the chambers aren't long enough for the .327 mag). I love it, and hunt small game with it fairly often, but factory ammo is very hard to find, and expensive. I handload, but haven't seen a single box of .32 bullets for loading in my local shops for over a year. Even online they can be hard to find. Handloading will help with available loads, but bullet selection will likely never reach that of the .357 caliber.
I plan to start casting my own pretty soon as a result. If you don't mind these limitations in available ammo and components, then maybe a blackhawk in .327 mag, or a Single Six in .32 H&R mag would suit your needs. That said, common, easy to find ammo is an advantage, and if it's also cheaper as a result of it's common availability, it's a win-win situation for most shooters that aren't looking for somethiong "different"
Which brings us right back to the .38 special/.357 mag. Ammo is widely available, and reloading components are fairly easy to find. In a pinch, you can load 9mm bullets in .38 cases with a heavy crimp to compensate for the .002 difference in diameter. I "aquired" a bunch of 115 grain 9mm hollow points (about 1000 of them), and since I don't shoot a 9mm, that's what I've been doing with them. They actually shoot pretty well from my .38 special revolver.
So, it seems to me that anything less than the .38, while being a "good choice", has limitations imposed by a lack of handgun and ammo choices. Anything bigger than the .357 will likely be more than you need for what you describe. The bigger cartridges cost more to shoot, regarless of whether you handload or use factory fodder, while not really offering any advantage for the uses you describe.
Pistols over .38 caliber are restricted, and not permitted over .45 .:barf: After I get the pistol then I must attend 6 competitions each year so I can keep the handgun ($) At this point I am still not allowed to "hunt" with a handgun.