Short answers, pulled from the story.
The Realm of New Zealand includes New Zealand, the Cook Islands, and Niue as three autonomous jurisdictions united under one crown. The Ross Dependency forms a constitutional part of New Zealand according to government definitions yet remains subject to international treaties.
Sir Tom Marsters has served as King's Representative in the Cook Islands since 2013 representing the monarch locally. This role acts de jure without subordination to the governor-general under the Cook Islands Constitution.
Dame Cindy Kiro took office as governor-general on the 21st of October 2021 following Dame Patsy Reddy's term ending on the 28th of September 2021. She represents the head of state across the entire realm taking on all dignities and reserve powers.
Only four other countries recognize New Zealand's sovereignty claim over this Antarctic sector between 160° east and 150° west longitude. Russia and the United States do not recognize this territorial claim leaving the matter unresolved alongside all other Antarctic assertions.
A 2016 poll indicated fifty-nine percent of the population supported transitioning from monarchy to republic with a resident head of state. Should such change occur the Ross Dependency and Tokelau would remain dependent territories while the Realm continues existing without New Zealand itself.