Was she born in Thailand? If so, you don't need to do anything in a hurry. She is legally in Thailand if she was born here. However in time you will need to get her a US passport as she will need this to leave Thailand. When she leaves You just show her new US passport and Thai birth certificate at passport control and she leaves. Should you wish to bring her back to Thailand you should get her a visa. If she has an O visa you can add her to your Extension as your dependent.
Yeah, that would be difficult. I returned to the UK and did it all from there, but we were planning to be there anyway. Maybe getting a UK based friend or relative to do all the posting would work. Either way, it's a pain that it all has to be done in the UK.
I can't speak for New Zealand, but we had to get our UK marriage certificate legalised and then certified in the UK before we could use it for getting dependent extensions. Hopefully your Embassy can do it here!