Is it a condition that a woman should untie her locks while making Ghusl (ritual bath)?
Ghusl from Janabah (ritual impurity), or menstruation obligates that water reaches the roots of the hair in order for the Ghusl to be valid, but if it doesn`t, then hair locks must be untied for water to reach them, and for Ghusl to become valid.
Does using a respiratory device (oxygen) affect the validity of fasting?
Using an oxygen device does not affect the validity of fasting because oxygen is a gas with no physical substance.
However, if medicinal substances with physical particles are added to the oxygen, its use would break the fast, as it enters the body cavity through a natural passage.
Can Zakah or Zakat al-Fitr be given to a poor brother?
Yes, it is allowed to give Zakah or Zakat al-Fitr to a poor brother, as he is considered among the eligible recipients.
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.