I am from perth, its lovely here, you are comming at the best time of year weather wise, although it will be very hot by the end of december.
Cloth nappies are no problem:http://nappyexpress.com.au/khxc/ccp0-catshow/hire.html
this is a nappy hire service that cleans them etc. i think there are a couple of others i remember looking at a couple before poppy was born, i can look for more if you want.
cost of living is quite high although we get by on next to nothing, you may have trouble finding a rental as they are in high competion as everyone wants to live here
public transport is fairly good within the metro area, if you have a consesion card it is cheap.
Beaches- the best in the world
really they are nice, although they will be busy at that time of year. other receration is pretty good, lots of sheap and free stuff to do, as you will be here for at least part of the school holidays there will be lots of kids activities. Oh and you will be here for the arts festival in northbridge, pride parade and a few other things going on.
Childcare can be difficult, i would suggest you pre book, if you are interested i can help you find the better childcare centres as i used to work in the industry, although all must meet certain requirments to be open.
Vax no issue i doubt you will be asked, childcare may ask for a vax record but that consists of you telling them none, basically its just so they can tell you if there are any outbreaks. No rampant VPDs that i can think of
I dont use sposies so cant help with price, but if you want i could go and look? there are plenty of laundromats and things though, and for that length stay it might even be worth buying some flat nappies here they are fairly cheap, and i know the best brand
Feel free to PM me if you need more help, and make sure you visit Freeo like the other posters suggested, it is the heart of alternative.