Can a woman perform I‘tikaf in her home?
No, a woman's I‘tikaf is not valid if performed at home, because her home is not a mosque.
How do I deal with whispers (waswasa) in ablution and purification?
Whispers in ablution are from Satan, and the Muslim should not pay attention to them. He is not required to re-perform ablution or repeat washing a limb because of waswasa. He should always base his purification on its being valid. It is recommended for him to frequently say "La ilaha illa Allah" (There is no god but Allah), because Satan recoils when Allah is mentioned. And Allah the Almighty knows best.
"I told my family at night: 'I will fast tomorrow as qaḍā' (a make-up fast),' but I forgot to make the intention before sleeping. Is my fast valid?"
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The intention is obligatory in acts of worship. For obligatory fasting, the required intention is to resolve to fast during the night [before dawn]. Its proper locus is the heart, while verbalizing the intention out loud in acts of worship is a sunnah — so that the tongue may align with the heart. If the intent behind the statement in question was the intention to fast, then it is valid; but if the intent was merely to inform someone, or reflected hesitation or doubt, then it is not valid.
It is stated in *Bushrā al-Karīm* (Vol.1/P.216): "It is a sunnah to verbalize the intention — that is, whatever has already been intended in the heart, whether an obligatory or a voluntary act — such as specifying the number of rak‘ahs, just before the opening takbīr — so that the tongue may assist the heart, and because this is further removed from whispered doubts (waswasah), and in order to accommodate the view of those who held it obligatory in every act of worship that requires an intention (even though this is a minority position), by analogy with Ḥajj."
Accordingly: if the intention to fast arose in your heart before the second adhān [i.e., before Fajr], then your fast is valid.
And Allah, the Exalted, knows best.
What is the ruling on reciting the Chapter after Al-Faatihah , and what should a worshiper who forgets it do?
All perfect praise be to Allah, The Lord of The Worlds. Any other part of Qur'an that a worshiper reads after Al-Faatihah during the first and the second Rak`ahs is a Sunnah, and forgetting to do so doesn`t nullify prayer, and doesn`t require performing Sujood As-Sahw (prostration of forgetfulness). And Allah Knows Best.
