Is it permissible to perform the eleven rak'ahs of the Qiyam al-Layl prayer in two stages: four rak'ahs after the 'Isha prayer, and the remaining (four or seven rak'ahs) an hour before the time for Fajr prayer begins, or to perform all of them before Fajr?

What you are doing is permissible because the time for supererogatory (nafila) voluntary prayer extends from after the Isha prayer until the Fajr prayer. It is better to delay the supererogatory prayer until the last part of the night for someone who knows they will wake up. Some of the early Muslims (Salaf) preferred to hasten the Witr prayer before sleeping, while others were of the opinion to delay it until the last part of the night. If someone prays Witr at the beginning of the night and then wakes up, they may perform voluntary prayers but should not repeat Witr, based on the Prophet Muhammad's (peace and blessings be upon him) saying: "There cannot be two Witrs in one night."