Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
Do pregnant and breastfeeding women have to fast?
Pregnant and breastfeeding women are required to fast. However, if fasting causes them harm or unusual hardship, they may break their fast but must make up for the missed days later.
If they break their fast solely out of fear for the fetus or the child, then they must both make up the fast and give fidyah (feeding a needy person for each missed day), as the benefit of breaking the fast was only for the child.
Is it permissible for a woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.
What is required of a traveler or a sick person if they broke their fast and then their excuse ceased?
If a traveler settles or a sick person recovers after having broken their fast, it is recommended for them to refrain from eating and drinking for the rest of the day, but it is not obligatory.