Is the fasting on days when the questioner felt water inadvertently enter her throat while rinsing her mouth, as well as the fasting on days when she suspected having an involuntary nocturnal emission (wet dream) without anything being discharged, and her prayers thereafter, valid, or must she make up those days?

The cure for waswasah (whisperings/obsessive thoughts) is to ignore them and not pay them any attention. There is no need to make up the days in which one doubts that water reached the throat, and giving in to these whisperings opens a door to great evil. Days are only to be made up if one is certain of having broken the fast due to menstruation. The unintentional reaching of some water to the throat does not invalidate the fast. Ghusl (definition) (ritual bath) is only necessary when one is certain of the emission of semen which necessitates it, and prayers or fasting are not to be made up without a valid reason. Making up obligatory fasts is not required immediately or consecutively; rather, it is permissible to spread them throughout the year, provided that they are made up before the following Ramadan (definition).