What is the ruling if someone wipes his head then shaves it - must he repeat it?
If a person performing ablution shaves his head after wiping it during ablution, he is not required to re-wipe his head or repeat the ablution. And Allah the Almighty knows best.
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 should someone do if they fasted for 30 days in their country, then traveled to a place where people are still fasting?
If a person completes 30 days of fasting in their country and then travels to another country where people are still fasting, they should continue fasting with them until they observe Eid. Even if they have already completed 30 days, he/she should abstain from eating and drinking and join the people of that country because he/she has now become part of that community.
What is the ruling of Islamic Law on leaving the Sunnah prayer of Dhuhr due to fatigue?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
According to the general rule, a Muslim should strive to maintain the Sunnah prayers before and after the obligatory prayers (al-sunan al-qabliyyah wa al-ba'diyyah). One is permitted to pray them sitting from the outset, but if one leaves them due to severe fatigue, there is no objection to doing so. It is recommended (mustahabb) to make up (qadā') a time-bound supererogatory prayer (nafl mu'aqqat) at any time if it was missed.
Al-Shirbīnī (may Allah have mercy on him) said: "If a time-bound supererogatory prayer is missed, it is recommended to make it up according to the more authoritative view (al-azhar), based on the hadith recorded in the two Ṣaḥīḥs: 'Whoever sleeps through a prayer or forgets it should pray it when he remembers it.' This is also supported by the fact that the Prophet (peace and blessings be upon him) made up the two rak'ahs of Fajr when he and his companions overslept in the valley and missed the dawn prayer until the sun had risen [narrated by Abū Dāwūd with an authentic chain]." And Allah, the Most High, knows best.