Is the aforementioned supplication, which begins with "I seek forgiveness from Allah, the Almighty, for every sin I have committed..." and ends with "I seek forgiveness from Allah, the Almighty, for myself, and for my parents, and for the believing men and women, and the Muslim men and women, living and dead," authentic and comprehensive?

There is no harm in the aforementioned wording of seeking forgiveness (Istighfar) and the meanings it contains. However, it is not authentically transmitted from the Prophet, peace and blessings be upon him, nor from any of his companions. It is better to seek forgiveness using the authentically transmitted wordings and to avoid elaboration.