Is it permissible for the woman to use her mobile phone while on the street?
All Perfect Praise is due to Allah, The Lord of The Worlds
Modesty should be observed by the woman at all times, so she is free to use her phone while on the street as long as it is done within the limits of Islamic manners. And Allah The Almigty Knows Best.
What should a person, who delayed making up missed fasts of last Ramadan due to the continuance of the same excuse, do?
It is permissible for one, whose excuse for not making up missed fasts of last Ramadhaan has continued, to delay fasting until the excuse ceases to exist, and he is considered neither sinful, nor obligated to pay a ransom. And Allah Knows Best.
Is it permissible for me to eat from the animal that I slaughtered for Allah to protect my family?
It is permissible to eat from the non-vowed animal sacrifice, and the person is rewarded based on the amount of meat that he had given to the poor. However, there is no evidence in Islamic Sharia indicating that slaughtering an animal protects one`s family, but it is a way for thanking Allah, The Almighty, for his grace.
Is it permissible for someone to make a supplication to Allah to marry an already married, or an engaged woman?
No, it isn`t, and the person who does so is sinful before Allah, The Almighty.