I have heard from a LOT of folks for years that Continental airlines are the best airlines for shipping pets.
I know Delta ships pets too but don't know anything else about Delta.

I know your dog will need to UTD on all vaccines and an exam & health certificate within 10 days of flying.

Sorry I can't be of any help with what carriers are airline approved. Your best bet would be to find an airline & ask them what they accept. Good luck and have fun!