Is it permissible to give Zakah (obligatory charity) to one`s poor sister, or brother?
It is permissible for one to give the Zakah to his poor sister, or brother if providing for them isn`t due on him, and the sister doesn`t receive sufficient provision from whomever that is due on. And Allah Knows Best.
Is a woman sinful if she fasts out of shyness from her family while menstruating or in postnatal bleeding?
It is forbidden for a woman who is menstruating or in postnatal bleeding to abstain from food and drink with the intention of fasting. If she fasts out of shyness, she is sinful, as her fast is not valid.
However, if she refrains from eating and drinking without intending to fast, it is not forbidden, but she unnecessarily burdens herself without need.
Is Zakah (obligatory charity) due on the Zakah money received by a poor person, and reached a Nissab (minimum amount liable for Zakah), and a whole lunar year had lapsed over having it in his possession?
Yes, the poor who possessed a Nissab for a whole lunar year is obliged to pay the Zakah due on that money even if it was given to him as a Zakah money in the first place. And Allah Knows Best.
Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.