Is a permissible ruqyah that includes Al-Fatiha, Al-Mu'awwidhat (Surahs Al-Falaq and An-Nas), Al-Ikhlas, Ayatul-Kursi, and other verses, for a duration of three hours, Islamically sanctioned? And is it permissible to recite it along with a recorded reciter or on the radio?

It is permissible for a Muslim to perform Ruqyah on themselves, and this is preferable. The Prophet, peace and blessings be upon him, used to perform Ruqyah on himself. He would recite the Mu'awwidhat upon himself and blow (gently). There is no harm in the duration of the Ruqyah, nor in listening to a recorded Ruqyah, or in reciting along with a recorded recitation. However, merely listening to a recorded Ruqyah or one from the radio is not sufficient as a substitute for Ruqyah.