Is it permissible for a Muslim woman to wear trousers at home?

All Perfect Praise is due to Allah, The Lord of The Worlds 

Is it permissible for the man to put on revealing clothes? Answer: It is impermissible for the man to put on revealing clothes that show his Awrah(private part) which is between his navel and knee, and exercising modesty is better. And Allah Knows Best.

What should a woman who is in a state of menstruation, or confinement do if she became ritually pure during the day of Ramadhaan ?

If she became pure during the day time of Ramadhaan, it is desirable for her to fast the rest of that day, and to make up for it along with the days that she had missed due to menstruation.

Is it permissible for us to sever ties of kinship if our blood-relatives` gatherings involve acts of sin?

Maintaining kinship ties is mandatory, thus if visiting your blood-relatives hinders their sinful acts , then you should do so. However, if their sinful acts persist while you are at their gathering, then it is sufficient that you maintain kinship ties via telephone and the like.

Is it permissible for a woman who is observing `Iddah (waiting period) due to the death of her husband to travel for performing Umrah (Minor Hajj)?

A woman observing `Iddah of a revocable (Rajee`) divorce isn`t allowed to travel for Umrah except with the consent of her husband.