Is the aforementioned Hadith about "Al-Kifl" who repented of his sins authentic, and is he the same as the Prophet Dhul-Kifl (peace be upon him) mentioned in the other tradition? And what is the authenticity of the second tradition regarding the story of Dhul-Kifl's naming?

The outward meaning of the Quran indicates the prophethood of Dhul-Kifl, due to his being mentioned alongside prophets.

The Hadith of "al-Kifl," which Al-Albani deemed weak, is not Dhul-Kifl mentioned in the Quran, and the wording of the Hadith is "al-Kifl" without any addition.

As for the second narration, its authenticity or weakness has not been ruled upon, and it was narrated by Ibn Jarir and Ibn Abi Hatim from Mujahid and other Salaf.