Hi there! So, our dog was a bit bigger, which meant she had to go underneath/cargo. It's my understanding that animals under 25lb's can go in the cabin. Is that an option? We seriously considered starving Daisy down to make that weight...
We ended up hiring a pet mover because it all got too confusing and stressful. I am very glad we did, because at the end she ended up shipping cargo after we had left the States and we would have been in bad shape without their help. That being said, I will never ship her cargo again. I would ship her as baggage, but as cargo it took her 3 full days of transit by herself, and I feel slightly lucky that she even made it. It sounds like your family is planning to travel with yours as checked baggage, which will be much better.
One thing with Korean Air that we found is that one of the planes (737, I think) that comes from Seoul to PP is not capable of taking pets underneath. Make sure this isn't the case for your itinerary. Daisy ended up flying (again, cargo) from DC--> Amsterdam--> BKK--> PP
As far as paperwork goes, it's a bit of a nightmare. You will definitely need:
-Health certificate signed by your vet, including proof of rabies vaccine.
-This must be done within a certain time to meet your States USDA requirement for their approval within 10 days of travel
-Crate that VERY SPECIFICALLY meets the airlines criteria. This can be different for every airline. Korean Air, if I remember correctly, is pretty clear on their website.
-Call the airlines ahead of time and go through it with them. Even if you feel like they give you assurances, be prepared for your dog and family to get to the airport and be told that there are lots of things they didnt tell you.
Shipping tips:
-Freeze one bowl of water that will go in the crate with your dog.
-If he/she doesnt go in the cabin with your family, attach a bag with some water and food to the crate. This doesnt at all mean that they will use them at all, or properly. Daisy had beans in her bowl and had clearly been out of water for a while, with some of the bottles we had provided still attached to the crate.
-Your dog will probably show up really stressed out, but will be ok!
-Make and have your family carry copies of any documents that are required, including USDA cert/rabies cert/etc.
-Just because someone along the way says that everything is in order, dont trust them and keep double checking.
Just thinking about all this is making me stressed out for her return trip! Ah! Let me know if you have any other questions along the way, hope this has been helpful!