What are the times when prayer is not permissible, and when does the time for Fajr prayer end if the Adhan is at 5:30 and the Iqama is at 5:55, knowing that prayer is "a book appointed" but some may miss it due to compelling circumstances?

The time for obligatory prayers (Fard) is fixed, and it is not permissible to offer them earlier or delay them unless there is a valid excuse, such as combining prayers (Jam‘), or due to sleep or forgetfulness. Whoever misses a prayer must make it up at any time, including the prohibited times (A اوقات النهي), because the prohibition applies specifically to supererogatory (Nafl) prayers. The prohibition for supererogatory prayers begins after the Fajr prayer until the sun has risen to the height of a spear. Delaying the prayer beyond its chosen time (Al-Waqt Al-Mukhtar) is forbidden unless there is a necessity, but delaying it from the very beginning of its chosen time is permissible.