Most schools quite happily take students all the way through the year. Of course it depends on if there is a waiting list or not.
Lane Cove is an excellent suburb in Sydney and although I dont know anything about the schools I would venture to say that the public schools would be good. It's an affluent area - parents well educated. Also there would be very expensive top quality private schools near there too - I think.
We are moving to the US - so all my questions are the other way around
good luck