Do yourself and everyone else on the airline a favor and wait until your child is old enough to walk on their own before you fly. Just because the airline allows it does not mean it's s good idea. It's terrifying for them and amazingly trying for all those surrounding you on the plane. Think it through carefully and be considerate of others in your choices.
Inside US, birth certificate & attestation from her father. International, passport & attestation ( and for France, livre de famille)
Are you coming from out of the US? If so I think you need a passport for her. If not a birth certificate if they ask is what you need . But just to be safe I would call the airline to make sure.
Yes, you can travel with her birth certificate and her dad written authorization.
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