Is it permissible for a wife to leave her house without obtaining the permission of her husband, and not to come back unless he divorces her co-wife?
It is forbidden for the wife to leave without obtaining her husband`s permission, and it is impermissible for her to ask him to divorce her co-wife as this inflicts harm on the latter and such an act is forbidden in Islamic law.
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.
Is wearing a cap permissible for boys, or girls?
All Perfect Praise is due to Allah, The Lord of The Worlds.
In principle, anything is permissible sol long as there is no Sharia evidence prohibiting it, thus wearing the cap by boys is permissible as long as it isn`t done in imitation of the disbelievers. Moreover, a Muslim woman should avoid all that which attracts the attention of men because whatever leads to the forbidden is forbidden. And Allah Knows Best.
What is the Ruling on Reciting the Prophetic Narration?
All Perfect Praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions.
Reciting the Prophetic narration is permissible as many scholars used to, since in principle, the rulings of recitation have to be observed while speaking Arabic language. And Allah Knows Best.