Is it obligatory for one who doubts whether what exited is semen or prostatic fluid to adopt the Hanbali school of thought, and is her prayer valid if she adopts the Shafi'i school of thought? Does air exiting the vagina after relieving oneself expel semen if there is no lust? Does thinking expel semen, or is it only something tangible? Is it obligatory to rub the vagina when prostatic fluid exits, or is washing sufficient?

You are not required to adhere to a specific madhhab (school of thought), and one afflicted with waswasah (obsessive whispering/doubts) should adopt the easiest opinions. If you doubt whether what has exited from you is manī (seminal fluid) or madhī (pre-seminal fluid), then ghusl (ritual bath) is not obligatory for you, and your prayers are valid, especially since you are afflicted with waswasah. Manī might be discharged due to thinking, but close the door to waswasah, for the default is that nothing has exited unless there is absolute certainty. Pressing on your thigh does not necessitate the exit of anything. You are not required to rub the private parts due to the exit of madhī; rather, it is washed just as urine is washed. The cure for waswasah is to ignore it and not pay attention to it.