M.,
I didn't even know about needing a note from my husband to fly out of the country with our child, so thanks. I have never needed one to travel within the US with our child without my husband.
As far as the carseat, I didn't use one on the plane with my 3 1/2 year old. He is average size and the seatbelt worked fine. Even at 2 1/2, when he was in a carseat on the plane, I had issues with him kicking the seat in front of him, but he was smaller and I had safety concerns. It also isn't always easy to lug the car seat on the plane and strap it in the seatbelt, especially if you are traveling alone with him.
You can easily check the carseat and I purchased a car seat travel case with wheels which has extra room for other items as well. If you travel on an airline that does charge per bag, as some of them are starting to do, they should not charge you extra for any of your baby/child gear (car seat, bedrail, stroller - which can be checked at the gate, etc.).
I agree that if you do decide to bring the carseat on the plane, ensure it is FAA approved. I read below about the booster seat option and I didn't think they were FAA approved for air travel. But if you are planning to purchase one anyway, a booster seat is lighter to bring along and check - ensure that your child meets the size requirements for either the backed or backless booster seat.
Also, some folks have recommended this: http://www.kidsflysafe.com, but I have heard some airlines may not approve (although the website says they should) and I thought it was expensive given the limited time I could use it for my three year old - but someone mentioned you can rent it from ebay.
Good luck!