Is it permissible for a worshipper to pray ten rak'ahs of Qiyam Al-Layl after Ishaa prayer, then sleep and be woken up to pray the single rak'ah of Witr in the last third of the night, or must the Qiyam Al-Layl prayers be connected to each other to pray eleven rak'ahs in the last third of the night?

You may pray whatever you wish after the 'Isha prayer, then delay Witr until just before dawn. This is recommended, based on the Prophet's (peace be upon him) saying: "Witr is a single rak'ah at the end of the night." It is not a condition for the night vigil (Qiyam al-Layl) to be continuous, though performing it continuously is better.