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 sick person, who is incapable of fasting, and who didn`t make up for missed fasting days, do?

One who has missed fasting days is obliged to make up for them, but if he/she wasn`t able to because of an incurable disease , or old age, then he/she has to pay a ransom which is feeding a needy person for each of the missed fasting days.

What is the ruling on the chatting between a young man and his girlfriend?

Private chatting between the two sexes entails religious violations, and leads to prohibited attachment.

Is a wife considered divorced if her husband refused to go to bed with her(for sexual intercourse) for two months?

A wife doesn`t get divorced no matter for how long her husband refuses to have sex with her, and she had better go to court.