Not necessarily. Iβm on AA operated by BA. Iβm in business/first class. I canNOT choose free seats. I can pay for them if I like. Free seats will be available at check in, 24 hours prior.
Not sure about the cost or if they accept charge cards, but a friend advised us to take the certified cabs outside the terminal, don't go with those drivers that offer their service inside the terminal. Those have no certifications and often overcharge.