Is waking up half an hour before the Fajr adhan considered part of the last third of the night, and is one sinful if they woke up two hours before that but did not get up to pray, relying on the alarm clock?

The time you get up half an hour before the Fajr adhan is part of the last third of the night. There is no sin upon you if you wake up two hours or more before it, then go back to sleep and do not wake up for the prayer relying on an alarm, because the time for prayer has not yet begun. It is disliked to sleep before the prayer once its time has entered, unless sleep overwhelms you and you cannot repel it, or you are quite certain that you will wake up and there will be enough time for prayer and purification.