How can the mouth, ear, or a cup be purified from impurity if the running water used in these cases will find a place to collect, thus becoming stagnant water that becomes impure from the impurity, which then necessitates the use of other water that in turn becomes impure, and so on?

These problems arise from indulging in the severe obsessive thoughts you are suffering from. Indulging an obsessive person only increases their obsession. You must stop obsessing and completely disregard it. The problem you mentioned is not valid, for the water that enters the mouth to cleanse it from gum blood will exit, and other water will enter until the blood is gone and purification is achieved. This is unlike tap water and bucket water, which do not remove impurity, and your analogy is incorrect.