Is it obligatory to delay the Fajr prayer by a third of an hour after the Adhan, such that it is impermissible for a woman to pray in her home except after the Iqamah? And is this ruling specific to Saudi Arabia or general?

Prayer times are known only through the cosmic signs that the Shariah has made evidence for their onset. The time for Fajr prayer is the appearance of the true dawn, whose light spreads clearly from right to left. Prayer is not valid before its prescribed time. It is from the Sunnah (definition) that the second Adhan (definition) for Fajr be after the appearance of the true dawn. A woman in her home should pray Fajr if she witnesses the appearance of the true dawn. If she is unable to do so, she should follow the reliable Imams of nearby mosques or rely on an accurate calendar.