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.
Should missed Sunnah prayer be made up?
Making up missed voluntary acts of worship is from Sunnah.
Is it permissible to fast on behalf of the dead who died owing missed fast to make up?
His relative should fast on his behalf, and it is permissible for the guardians to give permission to non-relatives of the dead to fast on his behalf as well. And Allah Knows Best.
What is the ruling of Sharia when a man releases his sperm out of his wife`s vagina?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
This is called coitus interruptus and it is permissible according to Sharia when practiced by agreement of both spouses. And Allah The Almighty Knows Best.