I'm pretty sure Kiwis can't vote in Aussie elections though. And they can be deported back to NZ.

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.