sounds like the question's been answered, but I thought I'd just get on to say that I've flown with my daughter's car seats several times, and never had any problems at all--if there were a problem, they could also replace it for you. When they take it, it's checked in separately from the luggage. It is kept in a separate area in the hold and is usually in a separate area when you land. You have to drop it at a different section of the airport or you leave it at the hangar.
Personally, I'd be more worried about it being shipped. It will get banged up all over during shipping and will be thrown around even more on their own flights without some knowing that it was someone's baby equipment and needed special care. It could easily get tied up in customs or be lost, and you wouldn't receive it in time for your journey and end up having to pack another one--unlike if you leave it right at the plane itself. I guess I don't really get the decision, because it will be banged up at shipping centres around various countries, in shipping trucks, and on their own airplanes. I think you'll get less banging around with one flight and no other travel to shipping offices (plus it's free). But, if you want to do it that way, I can't see any reason you couldn't ship it. They wouldn't look at that, but be prepared to prove that it is yours and previously owned, because they will likely charge you customs and VAT as others have described.