It's not quite that simple. A NZ citizen needs to be resident in Australia for a period of time before applying for Australian citizenship. But it is certainly a lot easier than many other countries.
And another feature of the system: if an Australian has any connection to NZ or knows where NZ is located, they’ll be deported to NZ if they turn out to be a criminal.
Not really. More like a permanent resident, which is still pretty nice. In the past they were closer to citizens (and many older NZers who come over can be grandfathered into these privileges to one extent or another, with some extra red tape), but that has more to do with the legacy of the Commonwealth than current agreements.