How to pay Zakah (obligatory charity) due on articles of merchandise?

Articles of merchandise are estimated by their whole sale market price at the end of each lunar year, and (2.5%) of their value is paid as Zakah whether it (value) went up, or down compared to actual purchasing price, and whether the increase (profit) was in the article itself such as an increase in the animal`s weight, or in the prices themselves. And Allah Knows Best.

Is it permissible for a pious Muslim woman to stay with her husband who had quit praying out of laziness?

Quitting prayer is one of the major grave sins that come after disbelieving in Allah, but the wife of such a person isn`t considered divorced, rather, she should exert all her efforts in order to bring him back to the way of Islam. However, if his sin is likely to lure her from the way of Islam, then she had better separate from him by lawful means such as Mokhal`aa (When a wife pays a compensation for her husband in return for divorcing her). Moreover, if she exercises patience, and remains steadfast on her faith, then there is no harm in doing that.

I`m a university student. While I was sitting for an exam, the professor caught my fellow student cheating. I heard him telling her to hide the dossier. After the exam, he asked me to testify that I saw her cheating although I heard him telling her to hide the dossier but didn`t see her cheating. What is the ruling of Sharia on this?

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

Wearing trousers at home isn't prohibited so long as non-Mahrams do not see her. And Allah Knows Best.