I have to go to work so I don't have time to research the seat at this moment. It sounds like you do not have a base for the seat? For easier installation and ease of use, you can look into buying a base that stays in the car and then you just click the seat right onto it. That eliminates the need for strapping the seat in with the seat belt every time you move baby.
I'll have to look for a manual, but it looks like you need a lap/shoulder belt to install the seat, therefore you probably won't be able to use it on a plane. I could be wrong. If I have time later today I'll do some digging and see what I can come up with.