Hi Ian,
Your choice of calibres is pretty good, but for the centrefire, whilst a .223 will do the job most of the time, I'd be leaning toward a .243 for the pigs and dogs. They'll do the same job but more effectively, particularly on a bigger animal. A .223 is ok if you're only spotlighting and/or you can get a shot at undisturbed game, but that's not always the case. Ammo and a good second hand rifle are generally easy to come by, and not too expensive. Recoil and noise is too bad either. As for brands, you're on the money with the European brands, but the Howa/CMC Mountaineer rifles are a good unit for the money also. A standard scope for such a rifle would be a 3-9 x 40 in a decent brand.
But any bolt action by a familiar brand would/should be fine.
As for you shotgun, don't buy a single barrel, it will kick like one of your daughters horses and wouldn't be pleasant to use. The same goes for most old side by sides as well.
You might have to spend a bit extra, but a second hand Miroku under and over will be worth every cent. It will be comfortable to shoot, especially recoil wise, and it will never wear out. You can usually pick them up for around and even under $1000. There's one on Used Guns at the moment for $550, with the transferring dealer Queensland Gun Exchange.
As for chokes, Half and Full or Three Quarter and Full are a common combination and would suit some pest eradication.
For ammo, for the bunnies, size 4 shot in 32gram-1-1/8 oz. would go ok, and for the pigs and dogs, start at BB and go up, but as you go up in size, you go down in pellet count. BB's are a good compromise for hitting power and pattern density, 36gram- 1-1/4oz of shot here. And maybe a packet of 28 gram 9's for snakes if need be.
Good luck and have fun!
Paul