When does the time for the 'aqīqah lapse and pass?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
If the guardian was financially capable (mūsir) during the majority of the postnatal period (nifās) — which is sixty days — then the obligation of the 'aqīqah remains upon him until the child reaches the age of maturity (bulūgh). Once the child reaches maturity, the demand falls away from the father and those like him. At that point, it becomes Sunnah for the child himself to perform the 'aqīqah on his own behalf.
However, if the guardian was financially incapable (mu'sir) during the postnatal period, and then became financially capable after its expiry — that is, after sixty days — the 'aqīqah is no longer required of him. And Allah Almighty knows best.
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.
What is the ruling on the fasting of a woman who has reached menopause if menstrual blood flows?
If a woman reaches the age of menopause (which is usually sixty-two) and her menses have ceased, then she sees blood after that, and its duration is not less than a day and a night (24 hours), it is menstruation (hayd). If it is less than a day and a night, she is considered as having non-menstrual vaginal bleeding (mustahada), so she fasts and prays. However, she must perform ablution for every obligatory prayer after its time enters, pray immediately, and be treated as a person with a continuous condition. There is no specific end limit for a woman's menstruation; it is possible as long as the woman is alive. And Allah the Almighty knows best.
Is it permissible to divide one mudd of fidyah between two people?
No, it is not permissible to divide one mudd between two people, because it would not count as a full meal for either of them.