C.C.
I would definitely purchase a seat for him, especially on such a long flight. We have traveled frequently since my son was 3 months old. There was one time when I didn't buy him a seat (a trip to New Orleans for a cruise) and I regretted it.
On the flights, just make sure you have plenty of water and activities. We have purchased a portable DVD player for our son and it has entertained him pretty well. Just make sure you get one with a long battery life.
We've also purchased a GoGo Kidz Travelmate from One Step Ahead. They hook on to your child's car seat and create a stoller - your child rides in his car seat and with this contraption the car seat has wheels. It is an awesome alternative to having a stroller AND carrying the car seat.
Happy travels.