oh god slug, I'm so sorry about your sister. I can't imagine.
I have done a LOT of longhaul with small children (9 - 14 hours) but never to aus or nz so others will have better advice than me on airlines, stopovers etc. 18 months is about the worst stage as they are completely mobile but short concentration span and can't really be reasoned with. which is why I'm going to suggest drugs! no, I don't suggest you keep her doped up for the whole journey, but think you should seriously consider giving her a couple of spoons of vallergan or phenergan so that she sleeps when she should, because otherwise they get so overexcited on the plane it can be a nightmare. and the KEY thing about this is to test drive your drug of choice BEFORE you get on the plane. for example, phenergan knocks out about 70% of children, and sends the other 30% hyperactive. and you don't want to discover your child is in the 30% once you are on the plane. If I was doing it I would try and do nightflight, day stopover to run around and burn off energy, followed by nightflight, and dose with vallergan during both night flights.
I'd also recommend, for an 18 month old -
crayons and paper
books
small packets of raisins
various stuffed animals that can be used to put on variety performances and silly voices
trucks/cars
several changes of clothes including pyjamas to change him/herinto before you try and get her to sleep so s/he knows it is night
5 more nappies than you think you may possibly need
some sweets that you normally would never give him/herfor moments of extreme desperation.
Also insist that you get a bulkhead seat, swallow your natural reluctance and tell them your sad family situation and request very politely if they could try and keep the seat next to you empty (so you get one for sluglet too). You could get your dp to do this for you over the phone in advance if you can't face doing it yourself. airlines are normally pretty good about this.
Advantage of bulkhead is that you can use the space on the floor as a bit of a confined play area for her. put down an airline blanket before you do and then it is easier to scoop up the toys.
Finally, pack a food bag with sandwiches, babybels, juice box, banana, biscuits etc as the childrens food on airplanes is usually terrible, full of sugar and colourings that will send them nuts, and more importantly it is never ready when you need it. there are times when your child is starving and misbehaving and you'd kill for a banana to give them and it is useless to be told the food service is in 20 minutes.
Also, obviously, if you can possibly go club, don't hesitate. To be honest you are more likely to get a spare seat beside you because no one will want to sit next to you and sluglet!!
And don't worry about other passengers, there are always billions of children on NZ flights, it goes with the territory.
Finally don't worry about passports but a good tip is to take a photocopy of your marriage certificate with you and book your ticket in your married name so you can both have the same names on the ticket. I've been doing that and travelling with children solo for 5 years and never had a problem. They have different nationality to me and my passport is in my maiden name as well.
Once again really sorry to hear about your sister, best of luck with everything and I'll be thinking of you.