Is it permissible for someone afflicted with obsessive doubts regarding ghusl (ritual bath) and intention, who spends a long time performing ghusl, to pray Jumu'ah (Friday prayer) with the Imam and then make up the Fajr (dawn) prayer after Jumu'ah, knowing that he did not pray Fajr before Jumu'ah prayer?

You must ignore the obsessive thoughts about ghusl (ritual bath) and pay no attention to them. Perform ghusl in a normal manner by pouring water over your body until you are reasonably sure it has covered everything. This is considered valid according to the Shafi'i school. These obsessive thoughts have caused you to miss the prayer time, so you must abandon them and strive against your soul to leave them.