Do I use the abbreviated name of a country for the country of publication?
No, you should write out the name of the country in full, e.g. New Zealand. For the United Kingdom, it is better to write out the name of the specific country if you know it, e.g. London, England rather than London, United Kingdom.
Do I put the Australian states in for place of publication?
No, you don’t. APA says for place of publication within the United States, it is city and state abbreviation, e.g. New York, NY or Los Angeles, CA. For everywhere else, it is city and country, e.g. Melbourne, Australia.
Do I have to write out the full name of the publisher?
No, you don’t. Leave out words like Publishers, Co., and Inc. Keep Books and Press.