What is the ruling on doubting the emission of pre-ejaculatory fluid (madhy) due to sexual thoughts and fantasies during ablution (wudu) and prayer (salat)? Is ignoring it correct? And how does one purify themselves and perform Umrah with an inability to control flatulence and frequent sexual fantasies that cause the emission of madhy?

You must strive to guard your thoughts and busy yourself with what benefits you in religious and worldly matters, ignoring whispers [waswas] and paying them no mind. What you are doing by not investigating is correct, so do not rule that madhiy (pre-seminal fluid) has exited except with absolute certainty.

As for the issue of stool, if the emission of wind or stool is continuous, such that it occupies all the prayer time, then your ruling is that of one afflicted with incontinence (sahib al-salas). You should perform ablution for the prayer after its time has begun and pray both the obligatory and supererogatory prayers with that ablution. However, if you find a period sufficient to perform the prayer with valid purity, then you must wait until you can pray during that time.