Is it permissible to shower in a shared bathroom where children urinate, and does pouring water over most of its floor purify it? And is it obligatory to wash drops of urine that fall on the feet before performing ablution for ritual purification (ghusl) or for prayer?

You must strive to combat these whisperings (waswas) and seek to rid yourself of them. Know that the default state of the bathroom floor is purity (tahara). Therefore, do not judge it to be impure unless you have absolute certainty. The flow of pure water over the floor after urine has fallen on it is sufficient to purify it. Hence, there is no harm in you performing ghusl in this bathroom. It is sufficient for you to pour water over the spot that you are absolutely certain is impure, and to act upon the default state of purity for what you are not certain about its impurity. As for the water that splashes back onto your feet during istinja (cleaning oneself after relieving oneself), it is not obligatory to wash it unless it has changed due to impurity.