What is the ruling on reciting various supplications, such as Ayatul-Kursi, the Mu'awwidhat, Tasbih, and Takbir, a specific number of times before sleep, and is this considered an innovation?

The sleep remembrances you mentioned—some are authentically transmitted, and some are not. Among the authentically transmitted remembrances are: reciting Surah Al-Ikhlas and Al-Mu'awwidhatayn (the two protective surahs) three times with spitting and wiping, Ayat Al-Kursi (the Verse of the Throne), the concluding verses of Surah Al-Baqarah, and glorifying, magnifying, and praising Allah with the well-known numbers. Anything other than that is not authentically transmitted for sleep, so it should not be believed in or adhered to. As for seeking forgiveness, it should be for all sins, not just grave ones, and it is best to stick to the authentically transmitted formulas.