I believe it's up to the individual airlines whether they'll let you take your dog in the cabin with you. Pets need to be kept in a sealed, hard carrier which must not be opened during flight. I guess the airlines have to consider their passengers' comfort first in case, for example, the dog has a tendency to bark for prolonged periods or might cry.
I don't know if you're aware of all the complicated procedures pets have to go through before they can travel to NZ and, once you arrive there, your pet must go into quarantine for a month so that would involve a separation. I have heard of such a thing as 'home confinement' where the dog is kept in the house for the period of the quarantine, but I don't know if this is still carried out. It would be worth asking. I think most of us worry dreadfully about the welfare of our pets when they are being transported such long distances, but the great majority of them are absolutely fine, even tiny ones, if they are in the care of a reputable shipper.
Many people use a pet shipper to handle their little loved ones, which does cost a lot of money (usually goes by weight) but takes a lot of the stress out of a situation that is stressful enough. This one,
IPTLAX , operates from the US and, even if you don't want to use a company like this, you may find some of the information contained in there might be helpful. Collating all the documents and participating in all the tests has to be very precise, especially with timing, so anyone shipping their pet needs to know exactly what's required well ahead of time (we're talking months here). Some people were not prepared and got caught out because they hadn't started the procedures in time. And you don't want to get to the airport and find that something isn't quite right and you must leave your pet behind.