This website might be helpful, but as always ---- double check everything by calling the airline. Here's the number for Delta:
800-221-1212. Airline Pet Rules Regulations for In Cabin or Cargo/Checked Baggage for your pet airplanes.
I would call Delta (and China Air, if you're flying with them) ASAP because Delta limits the number of dogs they will allow in the main cabin to two. You need to get a reservation ahead of time to make sure your dog can get on the plane. Also, I don't know which airline you are taking over the ocean, but generally that airline's policies take priority. In other words, if you're flying Delta from Boston to Chicago, then China Air from Chicago to Taiwan, you will have to follow China Air's pet policy (including fees). You should
not have to pay pet carry-on fees to both Delta and China Air as long as you booked the ticket from (example) Boston to Taiwan.
Also, I know American Airlines will only allow dogs in the plane for up to 12 hrs. That means you may need to arrange to fly to, say, Chicago or San Francisco before getting on a flight to Korea. Whatever you do, I'd get on the phone with Delta right away. It's always a good idea to write down the name of the person you spoke to and get confirmation codes for the reservation. Good luck!