Short answers, pulled from the story.
The Realm of New Zealand is the collection of states and territories over which the monarch of New Zealand is head of state. It includes New Zealand itself, the self-governing associated states of the Cook Islands and Niue, the dependent territory of Tokelau, and the Antarctic Ross Dependency. It is not a federation; each component sits in its own customs zone.
The Cook Islands and Niue are the two self-governing states in free association with New Zealand. Both became self-governing from 1965 and 1974 respectively, and their arrangements are defined by documents including the 1983 Exchange of Letters and the 2001 Joint Centenary Declaration. The New Zealand Parliament cannot pass legislation for either state without their advice and consent.
Since 2013, Sir Tom Marsters has served as the King's Representative to the Cook Islands. The position is appointed by the Cook Islands Government and is not subordinate to the governor-general of New Zealand, acting instead as the local representative of the monarch.
Yes. New Zealand administers the Ross Dependency, an Antarctic sector between 160 degrees east and 150 degrees west longitude. Britain took possession of this territory in 1923 and entrusted its administration to New Zealand. Neither Russia nor the United States recognises the claim, and the Antarctic Treaty, which New Zealand signed in 1959, leaves all Antarctic territorial disputes unresolved.
Tokelau is a non-self-governing territory as classified by the United Nations. It came under New Zealand control in 1925. In referendums held in 2006 and 2007, Tokelau's population did not reach the two-thirds majority needed to move to self-governing status equal to that of the Cook Islands and Niue.
If New Zealand became a republic, it would retain the Ross Dependency and Tokelau as dependent territories, and the Cook Islands and Niue would retain their free association with New Zealand. The main uncertainty concerns the continued allegiance of the Cook Islands and Niue to the monarch; three options are possible, including retaining the King or Queen as head of state, adopting the new republican head of state, or having their own heads of state. A 2016 poll found 59 per cent of New Zealanders supported a republic.