Is it permissible for my father to pray two Rak'ahs after 'Ishā' then pray Witr with one Rak'ah and sleep, then wake up to pray two or four Rak'ahs and pray Witr with one Rak'ah? And what is wrong with that, if anything?

What your father does by performing Witr twice in one night is wrong and forbidden, due to the Prophet's (peace be upon him) saying: "There are no two Witrs in one night." The Sunnah is to delay Witr until the end of the night if one is confident of being able to wake up, otherwise, one should perform Witr at the beginning of the night, due to his (peace be upon him) saying: "Make the last of your prayers at night Witr." If one performs Witr at the beginning of the night and then wakes up later, one should pray as many two-rak'ah units as they wish, and not perform Witr again.