It depends on the size of the hog, The size of the caliber, the weight of the pellet and if you can get in a good head shot.
Assuming you are using a .177 caliber rifle. (your stretching the limits of the rifle)
I would select a H&N Baracuda Match .177 Cal 10.65 grain pellet.
Don't get hug up on speed. A heavy pellet will fly slower than a lightweight pellet but will fly straighter and have more knock down power than a lightweight pellet.
Hopefully your going after really small hogs, because I think you are stretching the limits of you rifle if it's a .177 caliber air rifle. I recommend a larger caliber, but it's up to you.
I doubt it.
Rodents and small birds
Not likely. Air guns are not as accurate as air rifles
Yes, small game and rodents.
Rodents and small game.
Yes.......
Yes:
Depending on the size of the animal. Then yes it can with a good head shot. If the animal is too big it may not kill it, and then you have one very mad animal at you.
What is the condition of the winchester. Poor to Excellent? 800X or 800 XS?
Small game like squirrels, rabbits and rodents.
Yes a C9a can be used to kill small game. The kill ratio is up to you and how good a shot you are. The best shot is a head shot with hunting pellets for a quick kill.
You mean Air Rifle. Yes this is fine. Air rifles (Not AirSoft) usually fire between 500 FPS and 1200 FPS