Hi,
I would definitely bring it. We travelled with our 10 month old recently and used the strap mentioned below. It didn't occur to me to have her sit in the car seat when it was attached to the rolling bag though, we just used our carrier in the airport.
I have heard that the car seats you get with a rental car can be a crapshoot. I suspect that if you went with a reputable company they would probably be safe enough, but may not be very comfortable. So it would certainly be more convenient to just get one with a rental car (although you will be responsible for installing it). If you aren't planning to get a rental car I would definitely bring your car seat. It is absolutely not worth it to me to take the chance on riding in a car or cab with my baby not in a carseat. I am sure that 99.9% of the time it would be fine, but a little inconvenience is worth the piece of mind for me.
Gate checking the car seat was very easy, and if you can manage to travel without checking luggage I'd be very impressed!
Our daughter did fall asleep for a little while on our laps (which she never does at home). I was amazed at how few toys she actually needed on the plane (we had a whole arsenal) but I think it was because she was being entertained by her momma and poppa the whole time. Jet Blue is great because they have the tvs it the seats and since she doesn't watch tv at home it is novel and can buy you a little time.
I am not so good at traveling light and I am very particular about what she eats so I made sure that we had a weeks supply of the foods that were most important to me (her organic cereal, ground flax seed, etc stuff that would be hard to replace) and I had a two day supply of everything else.
They were good about baby food. You have to present it in a separate container to TSA and they will inspect it, but they never gave me a hard time. I had a lunch size cooler full of yogurt, juice, water, and food.
Nurse, give a bottle or pacifier at take off or landing to help with the ear pressure.
Try to stay relaxed so they don't pick up on your stresses.
Have a great time!