I took my ds to NZ aged 3. Definitely need a buggy for airport if nothing else!
Walking a lot at 3 is not the same as keeping up with adults in the heat or when tired. Especially when adults also carrying bag with snacks/drinks, swim stuff etc. Or is he expecting you to be the pack horse while he carries her...
How does your dp think he'll manage to carry her, carry bags and cases and handle tickets, buy snacks in shop, have a cup of coffee while she's sleeping etc at airport... Buggy is also useful for piling stuff on/in if dd doesn't want to use it. Every time you get up for some reason (need toilet, drinking coffee), or can't get a seat etc he will be standing holding dd. Might seem easy now, but what if it was 2 hours. Plus you can strap her in when she is awake and not have to keep worrying about where she's running off too in a busy airport...
You have to put the buggy through the xray, but they let you take it right up to the plane still. You then get it back as you get off. They did manage to break the plastic bar that held it closed, but only because someone tried to force the buggy up for me to put ds in.
Don't take too many toys on the plane, my ds barely used any of what I took and I had to keep ducking under the tray/seat to retrive stuff as it fell off his tray. The airline gave us a pack with pencils, magazine etc in anyhow. Check if your airline has those little tv's in the back of the seat in front, most do now but not all. They usually have a kids channel with cartoons running on it.
As for when you are there you'll need what you would on any long journey. Stuff to keep amused in the car/rv.
Hope you have a fab time, it's beautiful over there and you are going at the perfect time of year. In NZ try and go to the Argadome which your dd will love:
www.agrodome.co.nz/Agrodome/Sheep-Show_IDL=6_IDT=1055_ID=7900_.html
and also the kiwi fruit farm here:
www.kiwi360.com/
Be warned the kiwi karts go much faster than you would think. My ds thought they were cool, just like the rides at disneyland had been! He still remembers bit of it even now.