I always make the decision about whether to book a seat for my child based on the length of the flight. And hour or two - no seat needed (especially for a 9 month old who isn't too big or that mobile yet). Three or four hours (or with an older and bigger child) - definitely spring for the seat if you can afford it.

If you do book a seat, I'd recommend bringing a car seat on board, as a 9 month old won't exactly be stable for sitting on his/her own in that seat.

I've never rented a car in Europe, but in my experience in the US, I've used my own car seat with plenty of success. Just make sure you know how to install it in a variety of cars beforehand.