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.
What is the ruling of Islamic Law on sitting between the two prostrations, and what is the ruling on the dhikr recited during it?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
Sitting between the two prostrations (al-jalsah bayna al-sajdatayn) is one of the pillars (arkān) of the prayer, while the dhikr recited during it is Sunnah according to the Shāfi'ī scholars. It is recommended (mustahabb) to say what was narrated from Ibn 'Abbās (may Allah be pleased with him), who said: "The Messenger of Allah (peace and blessings be upon him) used to say between the two prostrations in the night prayer: 'Rabbi ighfir lī, warhamnī, wajburnī, warzuqnī, warfa'nī' (My Lord, forgive me, have mercy on me, make good my deficiencies, provide for me, and raise me)." If one were to say a different supplication instead, such as "Rabbi ighfir lī" (My Lord, forgive me), the prayer would still be valid.
If the worshipper omits this dhikr between the two prostrations, whether intentionally or out of forgetfulness, their prayer remains valid and nothing is required of them, though they miss the reward of following the Sunnah. Leaving it out does not necessitate the prostration of forgetfulness (sujūd al-sahw).
It is stated in Hāshiyat al-Bājūrī 'alā Sharḥ Ibn Qāsim 'alā Matn Abī Shujā' (Vol.1/P.298): "The eleventh — i.e., of the eighteen pillars — his statement: 'sitting between the two prostrations,' meaning even in a supererogatory (nafl) prayer... his statement: 'with the supplication that has been narrated concerning it,' meaning: 'Rabbi ighfir lī, warhamnī, wajburnī, warfa'nī, warzuqnī, wahdinī, wa 'āfinī' (My Lord, forgive me, have mercy on me, make good my deficiencies, raise me, provide for me, guide me, and grant me well-being). Al-Ghazālī added: 'wa'fu 'annī' (and pardon me). Al-Mutawallī also added: 'Rabbi hab lī qalban taqiyyan, naqiyyan min al-shirk, bariyyan lā kāfiran wa lā shaqiyyan' (My Lord, grant me a heart that is God-fearing, pure from associating partners with You, innocent, neither disbelieving nor wretched)."And Allah, the Most High, knows best.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
