I've done quite a bit of flying alone with both my kids when they were little. You can check the carseat. It counts as a piece of luggage. I got one of those bags for the carseat, and it was a big help in lugging it from baggage to wherever. If it is an infant carrier that attaches to your stroller, you may be able to put it in the overhead bin on the plane (check to see if it meets the allowable carry on size), but then you have to carry it from stroller, thru the aisles and to your seat (unless you have a nice flight attendant who is willing to help). The stroller though is a wonderful thing to bring if you can, and they will stow it under the plane for you after you roll it and the baby up to the plane. You might want to check with the airline, they may be willing to do the same with the car seat. Since you are not buying a seat for her, you would be unable to put it in a seat on the plane though.
As for diaper changing, some planes actually have changing tables in the bathroom. I usually changed her on my lap if just a pee diaper, and hoped for no poopies. And it's awkword, but it can be done in the cramped space of the bathroom - either next to the sink, in the sink, or on the toilet lid. Just bring a changing pad, and lots of wipes. In a pinch, you can put down the tray in front of you (you may get some strange looks, but who cares!) and lay her on that if she is small enough.
Good luck:)
Lucy