I’ve only flown Lufthansa and United when taking my dog along in the cabin. Both airlines allow carriers of up to 22″ x 14″ x 9″. And yes, the dog has to stay inside the carrier during the entire flight but usually no one says anything if you open the top of the bag. For a flight within the US most airlines charge something like $80-$100 for a pet in the cabin. Oh, and if you take your dog in the cabin you have to let the airline know in advance anyway because only a limited number of pets are allowed inside the cabin.