Is one who wakes up to the Fajr Adhan and deliberately goes back to sleep, without taking measures to wake up before the time expires, sinful? And is a Muslim sinful if he delays a prayer until its very last moment, i.e., just before the next one?

Sleeping after the prayer time has begun is not permissible unless one knows or predominantly believes that they will wake up before the prayer's time expires. Delaying the prayer within its prescribed time is permissible as long as it is performed before its time expires, though it forfeits the virtue of praying at the beginning of the time. However, it is not permissible to delay the Asr and Isha prayers until the time of necessity without a valid excuse, according to many scholars.