What are the fluids that exit the human body, other than blood, pus, and purulent discharge, that are considered impure by the Hanafis and Shafi'is if they exit due to an ailment or illness? And are tears, sweat, saliva, and nasal mucus considered impure in these cases? And what is the validity and strength of these opinions?

What is mentioned in the question is the relied-upon and most preponderant view among the Hanafis in general, as, according to them, one of the nullifiers of ablution is the exit of impurity from other than the two private parts. Some of them justified the nullification of ablution by the exit of tears from the eye, and the like, if it is due to an ailment, fearing it might be pus or discharge. Others interpreted it as being discolored by pus or discharge, otherwise not. While some scholars considered this to be based on two different narrations within the Hanafi madhhab. And according to the view that what exits from other than the two private parts due to an ailment is impure, a small amount of it is excused. This detailed differentiation is found only among the Hanafis. As for others, they generally state the purity of tears, sweat, and the like, without going into such detail. The default state of things is purity, so nothing is ruled impure except with proof. Among the liquid impurities that exit from a human are blood, urine, madhy (pre-seminal fluid), wady (thick white fluid after urination), and vomit. As for semen, the preponderant view is that it is pure.