Welcome to FT and the OW forum, grigriteam!

Let me get the discussion started with answers to some of your questions:

1. Yes, you will be charged the NZ price and it is lower than the US. Also note that there is a 10% off sale on at the moment.

2. Technically, there could be a "penalty", however there has never been a report here were anyone has actually been charged when not using the last segment(s).

3. Your itinerary doesn't work. You need to check whether there are actually flights between the cities you want to go to. Use this map http://www.innovata-llc.com/onw/default.asp?show=MAPto start planning.The problem is that for a lot of the places you want to go to you need to go to the hub cities like HKG etc so you need 2 flights, not one which eats into your total of 16.

What you propose looks like this:

AKL-BNE-MNL-HKG-BKK-HKG-KTM-HKG-DEL-AMM-IST-AMM-TLV-LHR-DAR-LHR-CPT-LHR-EZE-SCL-UIO-MIA-GUA-MIA-MEX-JFK-LAX-AKL

So this is 27 segments in total, you are only allowed 16. You are also having too many segments in Asia and too many entries into Europe.

So how to fix this? You need to drop some locations, pick locations with direct flights or do some of the locations as separate trips. Maybe drop one continent all together? Otherwise, try to use the segments for your long flights.

Here is an example:
AKL-BNE-MNL-HKG-DEL-AMM-IST-LHR-CPT surface segment to DAR-LHR-EZE-SCL-GYE-JFK-LAX-AKL

I have droped KTM, BKK, TLV, GUA and MEX. I have replaced Quito with Guayaquil as this makes it easier flight wise.
Since you are passing through AMM you can easily go to TLV from there so you really don't lose this stop. There are many other variations possible. Try using the OW online booking tool to come up with a plan.

4. You can book using AA when you are in NZ.