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 is the best charity to offer on behalf of the deceased? Is it giving food, reciting Quran, giving money or supplicating? What is the best charity to offer on behalf of dead father and dead husband? What is the best continuous charity to offer on behalf of the deceased?

Praise be to Allah the Lord of the Worlds. The deceased benefits from every righteous deed offered on his/her behalf, be that continuous charity, reciting Quran or a pious son praying for him/her. However, the best righteous deed is performing Haj and Omrah on their behalf especially if he/she hadn`t performed that ritual for it remains a debt on them. The evidence on this is that Ibn 'Abbas (Allah be pleased with them) reported: A man came to the Messenger of Allah (PBUH) and said: "Messenger of Allah, my mother has died (in a state) that she had to observe fasts of a month (of Ramadan). Should I complete (them) on her behalf? Thereupon he (the Holy Prophet) said: Would you not pay the debt if your mother had died (without paying it)? He said: Yes. He (the Holy Prophet) said: The debt of Allah deserves more that it should be paid."{Related by Muslim}. And Allah the Almighty knows best.

Is it disliked to clip one`s nails at night, and is it permissible to burn them afterwards?

It is desirable to bury the cut hair and the clipped nails. As for burning clipped nails, there is no evidence forbidding that , but it is better to suffice with the aforesaid ruling-burying cut hair and clipped nails-as it is based on a well known evidence.

What is the ruling on uncovering the eyes while wearing a Khimar?