To what extent is the supplication attributed to Al-Khidr, which begins with "O Allah, just as You were gentle in Your greatness without being among the gentle," and ends with "Verily, You are the Oft-Pardoning, the Most Merciful," authentic?

This supplication has not been reported from Al-Khidr (peace be upon him) in the revealed texts. Rather, these are stories without chains of transmission, mentioned by Al-Ghazali and others. It is not authentic due to the lack of a sound chain of transmission. Furthermore, the correct view is that Al-Khidr had passed away before the advent of the Prophet (peace and blessings of Allah be upon him). There is no harm in supplicating with it for whoever wishes, provided that it is not made a fixed regular recitation or believed to have a special virtue. The successful person is one who is content with what the Prophet (peace and blessings of Allah be upon him) guided to, in emulation of him, for the Prophetic supplications and remembrances are the best things a Muslim can seek.