Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?
Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.
If a woman becomes pure from menstruation before the Fajr Adhan in Ramadan, is she required to fast?
If a woman becomes pure (from menstrual period) before the Fajr Adhan, she must fast, as the impediment preventing her from fasting has been removed. The Sharia maxim in this regard states: "When the impediment is removed, the obligation returns."
She should then make the intention to fast before Fajr and perform ghusl (ritual purification) for prayer, whether before or after Fajr.
When is supplication (du‘a) more likely to be accepted: before or after breaking the fast in Ramadan?
Du‘a is accepted at all times, and this is part of Allah’s mercy and generosity toward His servants.
The Prophet ﷺ said: "No Muslim makes a supplication that does not involve sin or severing family ties except that Allah grants them one of three things: either He grants their supplication immediately, or He stores it for them in the Hereafter, or He averts from them an equivalent harm." [Narrated by Ahmad]
However, in Ramadan, du‘a is especially likely to be accepted shortly before breaking the fast.
The Prophet ﷺ said: "Three supplications are never rejected: the supplication of a fasting person until they break their fast, the supplication of a just leader, and the supplication of an oppressed person. Allah raises it above the clouds, opens the gates of heaven for it, and says: ‘By My might, I will surely grant you victory, even if after a while.’" [Narrated by Al-Tirmidhi]
Is it valid for the 'aqīqah to be performed using the newborn child's own wealth?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is not permissible for the guardian to perform the 'aqīqah using the newborn child's own wealth, because the 'aqīqah is a voluntary act of giving (tabarru'), and a guardian is prohibited from making voluntary expenditures from the child's wealth. Should he do so, he becomes financially liable for what he spent. And Allah Almighty knows best.