Don,
yep, we can have shotguns. No Pumps or Semi-autos (I think Politicians watch too many movies), but singles, doubles, bolt action, and lever action are all OK.
10 gauge an larger is illegal also.
Nope, you can keep them in your home, inside a safe that weighs 150kG, or is bolted to the structure of your house with a minimum of 4 1/2"x2.5" coachbolts.
You must either have a signed statutory declaration by a farmer that you are shooting vermin on his property, or be in a club and attend 4 shoots per year to be elligible for a licence...licences for clays will not allow you to shoot rabbits.
The authorites recently put a lot of effort into trying to take down Michael Diamond (Olympics gold medalist, and I beleive the ONLY reason that there are guns still in Australia). They couldn't get him, and he's now competing in Athens later this year.