When we've flown with our kids I always take a copy of their birth certificates but have never been asked to show them, by TSA or the airline. I think they would mainly just want it as proof of age in two situations: 1) to make sure the kiddo isn't too old to be a lap child and therefore fly for free, or 2) as proof of age if you purchased a seat at the "discounted" child rate (though at least on Southwest, the child rate is actually just a slight discount off the most expensive fare, so it's always been cheaper to buy adult Wanna Get Away seats for our kids). I suppose if TSA thought there was something weird going on they might ask to see the birth certificate, but since there is no photo attached to it I don't know how useful it would ever be to them as a form of ID.
As far as the car seat question, I already posted a novel on a post about a similar question : ) , but basically I can't imagine getting my kids to sit still in a lap on a plane for any length of time, so we always take the car seats onto the plane with us. We recently took them on a trip to PDX with 2 Diono Radians, which are pretty heavy, and while it wasn't fun we still plan on doing that later this year for our trip to DL. I did have the Diono backpack straps for the seats but they were still very heavy and cumbersome, so we will most likely bungee the seats onto one of those rolling travel frame things and roll them onto the plane that way. There are some car seats that are lighter and more recommended for travel, but I think the ones I've seen have lower height and weight limits so those may not work for you. We are only doing DL and won't be renting a car, so we were going back and forth between buying CARES harnesses and taking the DL Express bus to the hotel, or taking the seats on the plane and having DK Livery/Lansky's do the to/from airport transportation, but I think we've decided on Lansky's for a few different reasons. But if you're renting a car there and will need the car seat anyway, I would just take it on the plane. I've known a number of people whose car seats were lost or damaged while flying and I don't think I would want to deal with that on vacation, plus from what I've seen both with my kids and other kids while flying, everyone seems to do better and be more comfortable in their own seat. Good luck!