Is the water that splashes from flushing the toilet excused, or the water spray resulting from washing children after they relieve themselves, and is it considered pure or impure, especially with the presence of obsessive thoughts about purity?

What will free you from obsessive thoughts in this regard is knowing that the default state of water droplets is purity (ṭahāra), and their impurity (najāsa) is only ruled with absolute certainty. With doubt, the default state of purity is maintained; for certainty is not removed by doubt. So, do not busy yourself with purifying what you doubt its impurity. Rather, build upon the default state of purity and act upon it until you achieve absolute certainty to the contrary.