I would not bring the car seat on the plane, too bulky and heavy. I always carry my son in a sling when I travel- much easier.
I have found that buying two seats is a waste with two kids because one ends up wanting to be held anyway. I flew to Europe last year with my kids and just bought one seat. They swapped out and traded sitting on my lap and even playing together in one seat. And with a sling you have the child secured to yourself in the event of turbulence or a rough landing.
We always go to the dollar store and buy some new things to play with on the plane. It's cheap and you cna pull something new out of your bag each time they need a distraction. Also, scheduling a long flight around nap or bedtime is the BEST thing you can do for yourself. My kids get sleepy at take-off and usually get a good nap in.
Good luck. I have traveled alone extensively with my two kids and it has been relatively easy/pain-free. Depending on where you are going you may find that bringing a stroller is a waste of time and effort. We have traveled a lot in Europe and their streets/sidewalks are not usually stroller accesable so a sling/baby backpack is easier to deal with.
Enjoy your trip.