Must one investigate the emission of prostatic fluid (madhy) and wash the private parts upon recalling sexual intercourse or experiencing sexual thoughts, and what is the necessary time for such an investigation?

You are not required to go and check for the emission of anything after thinking about intercourse or doubting the emission of madhy (pre-seminal fluid). In fact, you should not do so unless the matter reaches a level of certainty or strong presumption. The cure for doubt is not to leave until a sound is heard or a smell is found, as stated in the prophetic tradition. By not going to check, you are frustrating Satan, who whispers to corrupt worship.