Is it permissible for a man to lie to his wife? Is it permissible for a man to abandon his wife to work in Saudi Arabia, without her approval, although he is wealthy?
All perfect praise be to Allah, the Lord of the Worlds; may His blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
The Messenger of Allah commanded men to treat women kindly, so marital problems should be solved through understanding, and whatever causes harm to a woman is forbidden. 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 permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
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.