ACC will cover accidents for everyone in NZ, even visitors and it includes accidents in the gym (been there, done that). For eligible children doctor/nurse appointments are free. Eligibility will depend on your residency status, so you will need to check that. Physio appointments are not free, although they are subsidised by ACC if treatment is necessary after an accident. If you need to see a specialist, it does help to have private insurance as there are waiting lists here and a shortage of specialists in the public system (probably less of a problem in Auckland).
Not sure if NZ Gymnastics has some kind of collective insurance. We do pay a quarterly levy of about NZ$25, but forgot what it is for. I am also not sure if it is dependent on the level or not.
Also, Auckland traffic is nuts, so make sure you choose to live somewhere close-ish to work and then find the nearest club. I have friends who spend close to two hours getting to work and two back, not much fun.