Is it permissible for a woman to conclude her marriage without the approval of her family?
No marriage is permissible without the presence of a guardian and two trust worthy witnesses, and the woman whose family had denied her right in getting married to a God-fearing, financially and physically capable man should go to the court.
Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
A woman has asked her husband for Khulu`, her gold, deferred portion of the dowry and furniture. Is she entitled to that, knowing that her husband doesn`t want to divorce her?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
Khulu`* can`t takes place save before a judge of Sharia and he has the authority to make the suitable decision in this matter. And Allah The Almighty Knows Best.
* Khulu' is when parties agree to separate by way of consent, usually upon terms such as the wife agreeing to repay her Mahr (dowry) to the husband upon him agreeing to grant Talaq. The iddah period (waiting time after a divorce) of a woman who seeks a khula, is one menstrual cycle or one month if she is post-menopause i.e. ceased menstruating. This is to ensure she is not pregnant. This differs from when a man gives a talaq; the iddah period is three cycles or three months.
Is it permissible for a Muslim woman to leave her house adorned and without wearing her proper Islamic attire?
A Muslim woman isn't permitted to leave her house adorned and without wearing proper Islamic attire. And Allah Knows Best.