Notice, it is not both of you who apply for this visa.. Your partner makes the application. She should look at the INZ office and fees finder
http://onlineservices.immigration.go...or.htm?level=1, which will tell her which office will receive applications for her geographical location.
So 2, definitely not. 1, see what the website says. 3 - well, some people would do it, but it can cause problems at the airport on arrival. A visitor's visa is granted for certain purposes only, and with the expectation that the visitor will be leaving to go to their home in another country at the end of it. Visa waiver countries aren't visa-FREE. It's just that people from those places don't have to apply for a visitor's visa in advance of travelling to NZ, and can fill in the application card on the plane just before they arrive. So she would have to give her reason for visiting - but she's not visiting, she's hoping to stay. This could make for awkward questions with the officials at the airport.
You should also be aware that for any partner-sponsored visa, INZ look most closely at the time immediately before the application. For the partner-sponsored work visa, they will ideally see at least three months' GOOD evidence that the partners are living together right then. All that has gone before does not necessarily count with them. So an application before you leave - she and you are living together - would be more impressive than one made after you have been apart for a month or more. YOU know you are a true couple. INZ don't know you. They could suspect that your relationship broke up and you left to go to NZ, but your partner decided to follow, and both of you made up a story to get her a visa.
Whenever you apply, be careful to keep collecting evidence all the time the processing is going on - INZ will ask for more before the visa is granted, to cover that time, making sure you are still together. In your case, with the necessary time apart, you should be careful to save records of your phone calls, emails, etc., keeping in touch during that time. See old threads for suggestions about evidence.
https://www.google.co.uk/#q=site:enz...tnership+proof