Is a woman`s neck an Awrah (private part)?
All Perfect Praise is due to The Lord of The Worlds.
It is an Awrah, and it is forbidden for her to reveal it before non-Mahrams (Marriageable men). And Allah Knows Best.
It was found out that a man divorced his wife for the third time in 2005, but they continued cohabiting as man and wife. However, when they realized that what they were doing was unlawful, they came the Iftaa` Department in 2009 and were told that they can`t be together. What is the position of Sharia on this couple?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
Her Iddah* period starts after the third divorce. As for the continued cohabitation, it is unlawful because being ignorant about the rulings of Sharia while in Dar Al-Islam* is no valid excuse. If a baby came as the fruit of this unlawful consummation of marriage, then a judge of Sharia has the final say in this regard because lineage is a serious matter. And Allah The Almighty Knows Best.
* ʿIddah: a specified period of time that must elapse before a Muslim widow or divorcee may legitimately remarry. The Qurʾān (Sura,2/verse,228) prescribes that a menstruating woman have three monthly periods before contracting a new marriage; the required delay for a nonmenstruating woman is three lunar months.
* Dar Al-Islam: designates a territory where Muslims are free to practice their religion, though this often implies the implementation of Islamic law, whereas Dar al-Harb represents those lands ruled by non-believers
Is it disliked to clip one`s nails at night, and to burn them afterwards?
It is recommended to bury hair and nails that have been removed and to conceal them in the ground. As for burning nails, we have not found any specific prohibition against it, but adhering to what is supported by evidence—burying the nails—is preferable. And Allah Knows Best.
Does Husband fasting while being impure for having sexual intercourse with his wife at night permissible?
Yes it is permissible for you to fast because impurity isn't among valid fasting conditions, but you are obliged to perform ritual impurity in order to catch up Fajr prayer at its due time. And Allah Knows Best.