.223/5.56 can be used for deer in a pinch, though it's definitely a marginal round for this; here are plenty of example of people doing it. Shot placement is very important for this. Military FMJ 5.56 will be a bitch to clean out of a deer, though, if it performs properly and fragments; a whole bunch of little chunks of lead scattered through a torn wound cavity. If used against prairie dog-sized prey, there won't be a whole lot left. This is the sort of damage you can expect from .223 on that size prey:

You won't see a practical difference for the most part between the 7.62x51, 7.62x54R, 7.62x63 (.30-06) or 7.92x57mm rounds for hunting. I'd let logistics determine this one. In your region, scavenging 7.62x51 and .30-06 will be slightly easier than 7.92x57 with 7.62x54R bringing up the rear.
Also, slugs do not need a rifled barrel in a shotgun. Many are aerodynamically stabilized (and work just fine from an improved cylinder or looser choke. The typical North American shotgun slug is the Foster type, which looks kind of like a hollow point that gets shot backwards. The hollow lowers the density of the rear of the slug and stabilizes it in flight. The other major type is the Brenneke slug, which attaches, typically, a plastic wad to the rear of the slug which also stabilizes it. Both work much like a badminton shuttlecock.