Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
If someone regularly fasts the six days of Shawwal, are they obligated to fast them every year?
A person is not obligated to fast the six days of Shawwal every year, even if they have made it a habit. However, someone who regularly performs a good deed should not abandon it as long as they are able.
What is the ruling on someone who prays but does not fast, without a valid excuse?
Fasting is one of the pillars of Islam, and a Muslim's faith is incomplete if they neglect the obligation of fasting while being physically capable of observing it.
What is the ruling on making up the regular *rawātib* (sunnah) prayers together with a missed obligatory prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
It is obligatory to make up (qaḍāʾ) a missed obligatory prayer (farḍ). As for the timed supererogatory prayers (al-sunan al-muʾaqqatah) or the regular rawātib prayers, it is recommended (mandūb) to make them up, even if a long time has passed. Ibn Ḥajar (may Allah have mercy on him) said in his Fatāwā: "It is established in our school that it is sunnah to make up the timed voluntary prayers, by night or by day, even if congregation is not prescribed for them, whether much time has passed or little."
Accordingly, it is recommended to make up the sunan together with the obligatory prayer. And Allah Almighty knows best.
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Note on terminology: Al-nawāfil al-muʾaqqatah (timed voluntary prayers) refers to supererogatory prayers tied to a specific time or cause, such as the rawātib (regular prayers before and after the obligatory ones), Duha, and the like. The ruling given here reflects the Shāfiʿī position.