I just flew two cats from CH to the US in the cabin. I used Swiss which allows the cats in the cabin but cost 250 bucks each (ouch). American Airlines does not allow cats in the cabin as far as I am aware. You should check other carriers though, I find Swiss the most flexible if you can believe that but they make take an arm or leg. The price of being in the cabin went up about three fold since 2003 when we flew to CH, I can't believe they charge so much to shove a cat under the front of your seat. You also have to reserve a space for the cat because most airlines only allow 2 or 3 cats onboard per flight. If we travel again with the cats, they are not going in the cabin, too expensive.

No quaratine on either side, coming into the US, you need to have a health certificate from the vet and updated rabies shots within 30 days? of traveling. Going into Switzerland if I recall was very similar but that was six years ago. The rabies shot is required and will be checked at customs, I am not sure if CH requires the good health certificate. In reality, the whole process was much easier than we expected on both sides.