Is hair extension permissible?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is impermissible for the man and the woman to apply a human`s hair as extension, but using artificial hair is permissible for the married woman after obtaining the approval of her husband so long as non-Mahrams don`t see her. And Allah Knows Best.
Sometimes I receive a message containing a picture of someone desecrating the Noble Quran. The sender asks recipients to send that message to their contacts as an act of defending the Quran. What should I do when receiving such messages?
Praise be to Allah, the Lord of the Worlds.
You don`t have to circulate that message, because this isn`t the proper way to defend the Quran. And Allah The Almighty Knows Best.
What is the ruling on a person who broke the fast a day in Ramadan due to hardship?
He/she is obliged to make up that day.
Is it permissible for a Muslim woman to wear trousers at home?
Wearing trousers at home isn't prohibited so long as non-Mahrams do not see her. And Allah Knows Best.