United will only accept Transatlantic pets onboard in the Cargo Hold. United will allow small domestic cats/dogs in cabin but you must be in Economy and seated by a window. No in cabin pets allowed in Business/1st Class, due to inadequate seat configurations for under seat storage[747/757/767/777].
Air France does accept in-cabin pets for crossing the pond with prior approval from the Telephone sales service. Dogs/cats weighing less than 6kg/13.2lbs (including transport bag or container).
Lufthansa also allows in cabin pets[Dogs/Cats/Ferrets] for 8kg/17.6lbs total for animal & carrier combined.
I do not know about the seating for AF and if there is only a specific class that you may bring your pet into, but Lufthansa allows your pet to accompany you in Any Class! Just make sure you call them first before you click on your fare online, and they will tell you if there is availability on your chosen flight[s] for your pet in cabin.