I would NEVER check my carseat.... it will get damaged. They don't care if its a carseat, suitcase, or an expensive somethign else, they toss it in and it gets thrown around just like any other piece of luggage. Getting checked on the airplane can pretty much equal a crash.
That being said, you will need a carseat when you get there, so bring your carseat, but bring it on the plane. Don't beg for a seat for it, unless you paid for one. Have them put it in the little closet by the front of the plane, don't check it.
My boys are 6 and 2 now, so we HAVE to purchase seats for them now, but we've bought them a seat 99.99% of the time even when they were babies, and brought thier seats on the plane. Why? Its just easier, and when you book your tickets, if you ask, alot of hte times you can get a discounted seat for your under 2 child.
When you check in, ask if there is an empty seat on the plane so you can be there and have extra room for your lap child, but don't count on it because you didnt' pay for a seat for her. But you should bring the seat and have them put it in the closet.... you will need a seat in san fran!