When do the prescribed times for each prayer end, and is that with the entry of the time for the next prayer, such that the time for the Asr prayer extends until the Adhan of Maghrib, and the time for the Isha prayer extends until the Adhan of Fajr?

Not every prayer's time ends with the entry of the time of the prayer that follows it. There is a preferred time (waqt al-ikhtiyar) and a time of necessity (waqt al-idhtirar) for each prayer, and it is not permissible to delay a prayer until the time of necessity except out of necessity.