It all depends on your airline whether you will be charged under your baggage allowance for a car seat. We have 6 "bags" allowance, so it was never a problem. Once my DD "outgrew" her car seat on the plane at around 3 and could reach the seat in front of her with her legs, we started checking our car seat in a special car seat bag. The bag was eaten by Southwest, but the car seat was fine inside. I now back a backless booster into our regular luggage.
Personally, I wouldn't gate check a car seat. First of all, it's a pain to schlep the car seat through security and to/from the gate. Secondly, I would be asking the baggage handlers to do extra work for me carrying my car seat and up down the stairs to the hold. It was bad enough when we gate checked the stroller!