When we flew from Akl we were able to take our stroller right up to the gate, they tag it at check in and then take it off you as you get on the plane. It should be waiting for you when you get off at the other end
I took our own cooled, boiled water in the bottles we needed and formula in a dispenser. They didn't even look at it going through security in Akl, they just said as long as you have a baby it's OK

We also had jars of baby food and dry snacks like crackers and cereal.
C drinks formula at room temp but the family next to us got their bottles heated on the plane. They will rinse bottles for you too but no facilities to sterilise so if take one more bottle than you think you will need.
Does your DS take a dummy? Use a dummy clip! So you don't have to keep picking it up off the floor. We gave C the dummy during takeoff/landing if it didn't coincide with a feed.
HTH

Enjoy your holiday!