Is it authentically established in the sound Sunnah that Al-Khidr (peace be upon him) offered condolences to the Companions (may Allah be pleased with them) on the passing of the Messenger of Allah (peace and blessings be upon him)?

There is no authentic proof from the Sunnah that Al-Khidr (peace be upon him) offered condolences to the Companions after the passing of the Prophet (peace and blessings be upon him). The Hadith mentioned in this regard is weak in its chain of narration, as stated by Ibn Kathir. The العلامة (scholar) Al-Shinqiti clarified that even if the Hadith were authentic, it does not necessarily mean that the one offering condolences was Al-Khidr. Rather, it could have been one of the believing jinn, and the statement of the Companions "they thought it was Al-Khidr" is not a conclusive proof, due to the possibility of a mistaken assumption.