Is a woman`s prayer considered invalid if non-Mahrams (marriageable men) saw her offering it?
A woman`s prayer isn`t invalidated if non-Mahrams saw her offering it, but she had better pray in isolation.
Is it permissible for the woman who is observing Iddah after the death of her husband to exchange calls with her relatives and husband`s family call?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
It is permissible for the woman who is observing Iddah* after death of husband to exchange calls with them; however, when she speaks to non-Mahrams , she shouldn`t be soft of speech and she should be straight to the point. This because Almighty Allah Says (What means): "O ye wives of the Prophet! Ye are not like any other women. If ye keep your duty (to Allah), then be not soft of speech, lest he in whose heart is a disease aspire (to you), but utter customary speech." [Al-Ahzaab/32]. And Allah The Almighty Knows Best.
[1] The iddah is a waiting period that a Muslim woman observes after the death of her husband or after a divorce. The Quran says: For those men who die amongst you and leave behind wives, they (the wives) must confine themselves (spend iddah) for four months and ten days.
Is it permissible for a financially capable woman to pay the Zakat to her sons or one of them?
It is permissible for a mother to give her children from zakat if they are eligible to receive it, even though she is obligated to provide for them. This is based on the saying of the Prophet Muhammad (PBUH) regarding Zainab, the wife of Abdullah Ibn Mas'ud (may Allah be pleased with them): "Your husband and your children are more deserving of your charity" [Al-Bukhari]. And Allah Knows Best.
What is the ruling on someone who prays Fajr late, after sunrise, knowing that they wake up on time but return to sleep and do not perform it within its time?
It is obligatory to perform the prayer within its prescribed time. If someone wakes up and knows how to act, they must perform it before the time ends. If they delay it, they are sinful and must make it up. And Allah Knows Best.