What is the ruling on the prayer of one who is certain that madhiy (pre-seminal fluid) has been discharged, but does not check for its discharge? And is it permissible for him to sprinkle his garment with water in winter despite the cold weather? And what is the ruling on his past prayers if his father had ordered him to ignore the madhiy?

It is not necessary to inspect for madhy (pre-seminal fluid) if there is doubt about its emission, unless its emission is confirmed. If its emission is confirmed, then the method of purification from it is consistent with what you have described you do.

As for the coldness of the water, it is not an excuse. Cold water in Egypt does not cause great hardship, and washing the testicles with cold water is medically more beneficial.

Doubt about the clothes being contaminated with impurities from the bathroom should not be entertained, because certainty is not removed by doubt. Thus, the certainty of purity remains until you are certain of impurity.

As for your prayers without purification from madhy due to your ignorance of its ruling, please refer to Fatwa (definition) no. 145215.

As for your father, if you did not describe it well to him, he might have thought it was Waswasah (obsessive doubt) or sweat, and he might be excused. However, if he issued a fatwa without knowledge, he has committed something unlawful.