New Zealand single teacher (mother) travelling with kids

Hello, I am a single mother travelling to work in China with my 2 children aged 6 and 8. They were born in Australia, have Australian birth certificates. They have recently obtained New Zealand passports and citizenship. They previously had Australian passports which have expired . They only have my name on their birth certificates. No father is named. Their surname is different to mine. My name is on their birth certificate.

I am a New Zealand citizen, with a New Zealand passport. They have been granted New Zealand citizenship through me. I also have Australian citizenship. I currently reside in Australia as do my children.

What issues may I face in relation to the children's birth certificate? (ONE PARENT, NOT MY SURNAME, or staying long term if I am successful in gaining a long-term contract)

What processes will have to follow ( upon receipt of invitation to visit school), in regards to documentation?

What do you advise in relation to my daughter who is 18 Jan 3rd? She may want to visit for extended periods . She was born in NZ, has a NZ passport, resides in NZ and also has Australian Citizenship. My children will be travelling either alone or accompanied by someone else who will bring them to me in China. What may this involve in either scenario ?

Thank you for your assistance