I notice that different doctors take different insurances. does the doctor, if he/she in his/her own clinic typically decide what insurance to accept?

Yes: The doctor (or medical group, or foundation) who owns the clinic gets to decide which insurance companies to be in-network with. Patients with other insurance coverage can usually still use their "out-of-network" benefits. Always try to buy better insurance (get PPO, but not HMO) so you will have out-of-network benefits, and so the doctor will have more time to visit with you.