oh I just remembered that I ALWAYS bring the car seat and gate check it. I've flown a lot with my babies. DD#1 has an insane amount of ffmiles
Even if you haven't paid for the extra seat, the airline will give it to you if they have the space, so bring the car seat to the gate in case they have the room, that way you can all get a little sleep. Sometimes they won't let you carry the baby in the sling for take off and landing, so it can also give your arms a rest with a wiggly toddler. Also, on the bulkhead seats, the arms don't raise, so if you're travelling with someone you know and you have the choice, I like the non-bulkhead seats better because you can spread out more in the seat area.
If you don't have a whole row to yourself, check-in again at the gate. Gate agents are usually more helpful, and the CAN block off an extra seat in your row if it isn't already taken so you'll have a better chance of getting the whole row. Just be nice. Then if you still don't have the whole row, when you get on the plane, ask the attendants if there's a possibility of finding a vacant row. They will often check the available seats at the end of the boarding process and offer them to people sitting in rows with children so you can have more room.
I would bring that b/c just in case, especially with this latest security increase. It never hurts to be over prepared. I once had to fax the b/c to DH when he forgot it and was flying alone with dd(on southwest, they check every single time). But a passport would serve the same purpose to prove your child's age if you have one. I brought dd#1's passport instead of the b/c on our last trip, which I did alone with 2 babes
: Don't try it!!