For self defense, I think that the .223 would be fine (you will doubtfully have problems with black bears beyond just being a pain in the ass at times, and it may be better to have the state remove them then relying on SSS if they're being a big nuisance).
As for hunting, I'd invest in a 30-30, or maybe a slug gun. Somewhat inexpensive, ammo is cheap (and those cheapy wal-mart shells have taken plenty of game, hell, I used to shoot em at 300yds on deer and yotes, but with a 30.06).
The 30-30 in a lever action will have good capacity, decent resale (compared to what you paid for it), and good energy. A .357 would be another choice since you already have one, redundancy is nice.
As far as cats, I wouldn't worry about them. If one wants to kill you, odds are it'll have you before you get to your AR.
Deer with a .223, its been done, make sure that you're up to the task and know when to pass up shots (importante!!!), and make sure that its legal.