I am looking at moving to Australia (possibly Melbourne) from New Zealand. How easy is it for a New Zealander to buy a house or apartment over there? Also, can you have a dog in your own apartment?

You would be best to contact the banks and real estate companies for more information about mortgages and the purchasing process in aussie versus NZ. Having pets in apartments would depend on the individual body corp of each apartment block. some will and some won't allow pets.