We traveled with our DD when she was 3 months and again at 1yr. We were never asked for ID for her. We traveled with her on our lap, which is what I would reccommend. In fact, both times (round trip) we traveled with her as a lap baby, we were given a seat for her (at no charge) because the plane wasn't full. We were told that happens quite often. If they have 3 seats across, they put you on the two outside seats, and then unless the plane is full, no one wants to take the middle seat, which is left open to you. We ended up using the seat just to store stuff, and held her the whole plane ride. We just checked her carseat at the gate. Worked out great for us, and gave us more money to spend on souviners.
Our DD has never been one to sit in a carseat for long periods of time, so it kept her quite happy to be able to move around a little and switch from mommy to daddy.