Is it permissible for the mother to give her children from the Zakah (obligatory charity) of her money?
It is permissible for the mother to give her children from the Zakah of her money if they are Zakah-eligible recipients, and she isn`t obliged to provide for them.
What should the one performing the slaughter say at the time of slaughtering the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah to say the following supplication at the time of slaughtering the 'aqīqah:
"Bismillāhi wallāhu Akbar, Allāhumma laka wa ilayka, hādhihi 'aqīqatu fulān."
("In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of [the child's name].")
This is based on what al-Bayhaqī narrated in Al-Sunan al-Kubrā (Vol.9/P.511), that the Prophet ﷺ said: "Slaughter it in his name and say: In the name of Allah, and Allah is the Greatest. O Allah, this is for You and unto You — this is the 'aqīqah of so-and-so."
And Allah Almighty knows best.
Does using a gargling medicine break the fast?
If the medicine reaches the body cavity (jauf), the fast is invalidated. However, if it does not enter the body cavity, the fast remains valid.
Therefore, it is advisable to avoid using it during the day in Ramadan.
What is the ruling on a woman who gets her menstrual period while fasting?
If a woman gets her menstrual period while fasting, her fast (of that day) becomes invalid, and she must make up for the missed days after Ramadan. Allah has granted ease to women in this situation, and she will be rewarded for not fasting because she is following Allah’s command.