Hi,
Welcome to ShotgunWorld!! Stick around and register! You've just found the one stop answer to all your questions.
What can you hunt with a singleshot 12ga.? Anything you can hunt with a $50,000 custom made double. You will just get only one shot instead of two or more.
Many, many people here started out with a single shot. And many people still do. They are generally inexpensive (as you just found out) and tend to be simple to operate and are pretty rugged.
If the gun has no choke, that's called a cylinderbore choke. There's no taper at the muzzle to constrict the shotcharge. I would think for partidge and rabbits that should a fine choke to use. Your effective pattern should be good to about 20yds maybe a bit more. As for being too hard on your game, that's more a function of shot size rather than what choke you use. Me, I would use a 2 3/4" shell with an 1 1/8oz or 1 1/4oz shot charge of #6's for rabbits, and maybe #7 1/2's or 8's for partridge. They are pretty soft targets and it doesn't take much to kill either.
Your 12ga is generally considered an all around gauge. It will work of everthing you might want to hunt. The other guages, i.e. - 20ga, 16ga, 10ga, 28ga, and the .410 all have advantages/disadvantages for shooting/hunting. But the 12ga. can pretty much do everything you will ever need.
HTH,
Dale