I have a cat and no way would I fly with her. She is like a very needy child.
If I were to fly with her and she was on my lap, it would be a contest as to who could take the dog’s eyes first. She apparently dislikes all living creatures.
Housecats can be contained in the passenger cabin under the seat in a travel carrier for a fee. You put a fluffy towel under the carrier to smooth out the bumpiness.
I owned a small macaw that could fit in one and we flew from Austin to West Palm Beach without a hitch.
The animals may get noisy if fearful but talking to them in a soothing voice will keep them calm enough until you land.
In no way should an animal be in a passenger's lap, ESPECIALLY a dog. They belong in a carrier under the seat or in cargo or they shouldn't fly.