Is "Bismillāhir-Raḥmānir-Raḥīm" a verse (āyah) of Sūrat al-Fātiḥah?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The basmalah ("Bismillāhir-Raḥmānir-Raḥīm") is, without question, a verse (āyah) of the Noble Qur'an, and according to the Shāfi‘ī school, it constitutes a verse of Sūrat al-Fātiḥah itself. Al-Fātiḥah is a pillar of the prayer, and it consists of seven verses, of which the basmalah is one. The prayer is not valid without reciting it, nor if any of the Fātiḥah's verses, letters, or shaddahs (consonant doublings) are omitted or altered.
It is stated in Bushrā al-Karīm (Vol.1/p.202), from the books of the Shāfi‘ī school: "The basmalah is a complete verse of it [i.e., of al-Fātiḥah] in practice — and it suffices, for this to be established, that there be strong probability (ẓann), especially given that this has come very close to certainty (yaqīn), due to the consensus (ijmā‘) of the Companions on its inclusion in the muṣ-ḥaf [the written text of the Qur'an] in its script, despite their extreme diligence in excluding from it anything that did not genuinely belong to it — to the point that they even [carefully regulated] its diacritical dots and vowel markings." And Allah, the Exalted, knows best.
Is the marriage, which lacks a valid legal contract, a guardian, and a court registration, valid?
It is incumbent that a valid marriage contract be concluded in the presence of a guardian and two trustful witnesses, and it should be registered in the court to protect the rights of the wife. Actually, a valid marriage contract is what differentiates between sound marriage and fornication.
Some online stores offer a discount on a customer's first purchase — what is the ruling on creating a new account each time a customer makes a purchase in order to obtain that discount?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
It is obligatory upon a Muslim to uphold honesty in both his words and his actions. Allah the Almighty says {what means}: "O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know." [Al-Anfāl/ 27] And He says {what means}: "O you who have believed, fear Allah and be with the truthful." [Al-Tawbah/ 119]. And the Messenger of Allah ﷺ said: "Whoever deceives us is not of us." (Reported by al-Bukhārī.)
As for creating a new account for the same buyer, the terms and conditions of the website must be observed and respected. If doing so violates those terms, it is not permissible. And Allah the Almighty knows best.
A woman made a wig for herself from her own [previously shed] hair, which she wears on her head after having lost her hair [due to hair loss]. Is this permissible?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Since the hair belongs to the woman herself, the wig can be removed during the obligatory ghusl (ritual bathing), and part of the head can still be wiped during wuḍū' (ablution), there is no sharī‘ah objection to making this wig — provided she does not appear wearing it in front of non-maḥram men (Any males outside a woman's immediate family whom she is permitted to marry under Islamic law), since it is not permissible for them to look at a woman's detached hair.
It is stated in Rawḍat aṭ-Ṭālibīn wa ‘Umdat al-Muftīn: "Whatever it is not permissible to look at while still attached [to the body] — such as the male genitalia, a free woman's forearm, the hair of her head, and a man's pubic hair, and similar matters — it remains unlawful to look at even after it has been detached [from the body], according to the more correct position (al-aṣaḥḥ)." And Allah, the Exalted, knows best.