Ah, no and no. You think this is going to be easy? We're giving away a pilot certificate for Pete's sake! And because you're Kiwi it's even better. Your CAA has a form to fill out to send to the FAA.
Airmen Certification ? Verify the Authenticity of a Foreign License, Rating, or Medical Certification
Normally you send your paperwork to the FAA and they send you a letter back saying they've confirmed your license and show up at the FAA office you picked in the next 6 months. A handful of Commonwealth countries add the extra step of having the pilot send a form to their CAA.
The form for the New Zealand CAA and the FAA form are on the last page of the above link. And there is a contact email address for the FAA