try looking here
http://www.realestate.co.nz/rental/s...arch=furnished
Now these are furnished and on the North Shore but the majority of these will not accept pets, one accepts cats but not dogs.Also most of these tie you down for 6 months.
You may have to make a compromise somewhere. You have more choice if you can possibly hire furniture until yours gets here alternatively and this is only a suggestion. Is there any possibility you can leave your pets with family until you get over here then you can find somewhere when you are here that will accommodate for your pets.I know it may not be possible but its only an idea.