How is the cessation of madhy known after cleansing from it? And what is the ruling when there are transparent secretions, and it is not known whether they are madhy or pure vaginal discharge, after cleansing? And is istinja’ sufficient, or is examination required thereafter?

It is sufficient for purifying madhy (prostatic fluid) to pour water on the contaminated spot until it is highly probable that the impurity has been removed. The normal discharges known as vaginal secretions are pure but nullify wudu' (ablution). If you are unsure whether the discharge is madhy or a secretion, you may assign it the ruling you prefer. And avoid obsessive doubts (waswas).