Is the person afflicted with obsessive-compulsive disorder (waswas) pardoned, and what is the ruling on someone upon whom unknown drops splash while washing the private parts, and does dry impurity transfer, and is it obligatory to re-perform three months' worth of prayers for someone who did not perform ghusl (ritual purification) from menstruation for that period?

We ask Allah to avert evil from you and guide you. The examples mentioned are not errors; rather, they fall under the principle that the default state of things is purity, and this is a remedy for obsessive thoughts (waswas). It is permissible for someone afflicted with waswas to adopt the easiest and most lenient opinions due to their need, and this is a permissible dispensation. As for the ruling on prayers performed without a ghusl (ritual bath) after menstruation, there is a difference of opinion among scholars; you may follow Sheikh Al-Islam Ibn Taymiyyah in the view that making up these prayers is not obligatory. However, if you wish to make them up as a precaution, that is good. You can refer to the mentioned fatwas for how to make them up.