P.R.
I'm not responding to the what to pack, because it sounds like you got plenty of responses to that. I don't know if it's the age of the child (my son was 6 months on our first airline trip) or just the airline, but we almost didn't get to fly because I didn't have proof for the airline (not for TSA) to take him. I guess it may also depends if you paid for a seat for them. I would look at the airline's website and see what it says. A quick look at one airline says "proof of age" required for all ages. So, we were standing at check in without a birth certificate or any other document to prove he was ours, and phoning our doctor's office, begging them to fax something to the airline that said he was a patient in their office and when he was born. So, as soon as we got back home, I took our son to the DMV and got a state ID for him so we never have to go through that again. I imagine part of it is to prevent kidnappings, but I can tell you I did not need the added stress to get his boarding pass.
I just saw that you said Continental (https://www.continental.com/web/en-US/content/travel/airp..., so it looks like you don't need ID with them.
I would recommend a backpack as a carryon and/or a backpack or wrap carrier for the 2yo. Then you can have hands free for other things. I think at that age, I would use a Chunei carrier. (http://www.hosausa.com/) You put it on like a vest, and there is a seat built into it so that he feels secure while sitting in it. Both my husband and I used this on our last vacation with our 3yo. (http://www.ihosacanada.com/chuneireviews.php)
You are not required to have a carseat on an airline. They can just sit like everyone else. But, if you want to have a carseat, on the plane (opposed to checking it is as part of your luggage) then it has to be FDA approved. (In the past we checked a carseat, and they had plastic bags to put it in when we checked our bags.) The Sunshine kids seats are FDA approved, but very heavy. So, I bought a backpack carrying case to go with it, and my husband will wear it backpack style. I'm sorry I can't tell you if it's a good idea or not, since our trip with this seat will not be till the end of the month.
Feel free to pm me if you have other questions. Hope you have a good trip!