I would suggest you stay a mile away from the Ruger 10/22. Awful trigger and very inconsistant accuracy. I can never understand how they get such praise. Ok so there are loads of accessories and after market triggers but who wants to buy something that needs so much upgrading.
My advise is go for a Bolt action every time and first choice is a CZ with "set trigger" or a Ruger 77 all weather stainless. Use a 3-12X scope with a very fine reticle , .22 subsonics bullets and a silencer and you can shoot small critters discretely with head shots every time, out to about 75 yards. Trust me zeroing past that is a waste of time. Better to get just get used to your scope and remember the "hold over" required at various distances i.e. 100yards.
The .223 is a perfect choice for the small to middle sized critters and id strongly recommed a Remington 700 VSSF or TIKA (great trigger) but to be honest a .233 just wrecks Rabbits if you like eating them. Truth is i think you actually need two Rifles in both calibres cos anything else is a bit like trying to play Golf with only one type of club( not that i play golf). Best of luck with your choice and happy shooting Ken (Ireland)