Have flown with one several times and it has never been looked at, at check-in. You can keep it with you right up until you board the plane and they take it off you and put it in the hold. Same with car seats except you take them to a place for 'over-sized' luggage to go on plane (not included in your luggage allowance). May be worth checking with your airline though. Have a good trip!
When I flew with Iberia, they wanted me leave the pushchair with them at check-in but I as child was only 6monthd and she needed to sleep, they were very annoyed but I stood firm, they wrenched it from me at the boarding gate. In all the airlines I have used, I have never been allowed to take it on the plane myself.
I can say with near certainty that no airline will charge you for a push chair. Take a look at the airline you are (or are plannning to fly withs) policy online. It will be there under "baggage allowance".
I always take ours right up to the gate - then at least its got less chance of getting lost - its only got a few metres to go!