When is the time for the 'Qiyam al-Layl' prayer, and is it permissible to perform it in segments before and after the 'Isha prayer and consider it 'Qiyam al-Layl'?

It is not a condition for Qiyam al-Layl (night prayer) to be late or after waking up from sleep. Rather, its time begins after the Maghrib prayer, and prayer between Maghrib and Isha is considered part of Qiyam al-Layl, even though prayer in the last third of the night is better.