Is it permissible for a Christian man to embrace Islam in secret and is it permissible for a Muslim woman to marry him?
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.
His embracing Islam in secret is between him and Allah; however, it isn`t permissible for a Muslim woman to marry him except after he declares his Islam and it is ascertained that he is sincere about that. And Allah The Almighty Knows Best.
Is it permissible to offer Tahajjud prayer after Witr prayer?
It is permissible to offer voluntary prayer after Witr (An odd number prayer performed between Isha and Fajr) prayer. However, it is desirable that the night prayer be ended with Witr. Therefore, it is desirable for one, who thought that he was most probably going to wake up at night in order to offer Tahajjud, to offer Witr after it, but one, who fears that he might not wake up, should offer Witr before going to bed.
Is it permissible for a person who has to make up for missed fasts to fast six days of Shawwal (the month directly after Ramadan) before making up for the missed fasts?
Yes, it is permissible for him/her to do so, but it is better that they make up for missed fasts first. And Allah Knows Best.
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.