What is the ruling on the prayer of one whose shoe has been splattered with some urine, and must he wash it every time? And is the prayer valid while wearing a ritually impure shoe?

We advise you to turn away from obsessive doubts and not to give in to them. Know that the default state of things is purity, so you are not required to wash your shoe unless you are certain that it has been contaminated with impurity. In that case, wash it if you intend to pray in it.