Does not performing the Fajr prayer on time constitute disbelief, and does it necessitate performing Ghusl for the next prayer?

Not praying Fajr on time, if it is not due to intentional neglect or laziness, does not make a Muslim a disbeliever. If one takes the necessary steps to wake up and does not show negligence, then they are not sinful, for the "pen is lifted from the sleeper." To wake up, it is recommended to seek help from Allah, adhere to the guidance of the Prophet (peace be upon him) in disliking staying up late after 'Isha, and seek assistance from someone who can wake you up.