You'll be able to find answers to all your Qs if you take the trouble to access the FAA website.. I believe that it's
www.faa.org it's resonably user friendly. Also you can find a wealth of information on the Pprune seach facility as the subject has been covered
ad nauseaum over the years.
Assuming that you qualify on hours, you will have to sit a 'writtten' (online), you may be able to do this while still in NZ... it's very simple, little more than a PPL written in Europe... You will then have to take a flying exam which involves an oral followed by the actual flying... The oral can be anything from 2 hours upwards, depending on your examiner and your performance and goes someway to mitigating the ease of the 'written' exam... the flying exam can be be tough, again depending on your examiner, unlike the European (and probably NZ) flight tests, it combines the Instrument flying and general handling into one test, again, apprx 2 1/2 hours.
I'm not current on who to recommmend.. there are very many schools scattered country-wide though few will know exactly what's required by way of the actual flight test for the ATP as most US pilots simply do the CPL and wait until a company sponsors them for a type-rating, combining the APT check-ride with the type check-ride. You'll need to do some research.