Hi,
We always took a car seat on the plane until my ds turned four. Some people think it's a big hassle, others think it's worth the effort.
The FAA strongly recommends all children under 40lbs ride in a car seat on board planes. I would say it's not that big of a deal to carry a CRS through the airport if you have a strategy and plan ahead. We still used a stroller at that age, so we would just either throw the seat on the stroller and ds would walk, alternatively, you could also bungee it to a metal luggage cart to wheel it through the airport. The big tip I learned here for getting it on the airplane was to hold the seat base up so the widest part is on top - it makes carrying it down the narrow airplane aisle very easy.
When you get to the seat, the car seat should go by the window, and others have reported that if there are three people in the same party sitting in a row, then it's ok to put a f/fing seat in the middle if you want (it just can't block the exit of another pax in an emergency). You would install it the same way you do in a car with a lap-only belt, but on airplanes, the easiest way to install a f/fing seat by reclining the airplane seat all the way first, then thread the lap belt thru the fwd facing belt path of the car seat, buckle and tighten as much as possible. Then you have to put the airplane seat upright, which will tighten the car seat up the rest of the way. Oh, and if the airplane buckle latch plate is facing the plastic shell of the car seat, flip the latchplate over once before buckling and tightening, otherwise, it may be hard to undo (only a problem on certain car seats, like Britax for example).
hth and hope you have a great trip. btw we are in WA too.