Is it permissible for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
Is it permissible to trim the eyebrows if they become excessively long?
Trimming eyebrows is permissible if they become excessively long, yet; it is reprehensible. And Allah Knows Best.
Is it permissible for the big brother to force his sisters to wear Hijjab (Islamic wear) if their father didn`t do his part in this regard ?
All Perfect Praise is due to Allah, The Lord of The Worlds
The father`s attention should be drawn gently and politely, and the sisters should also be reminded with good words since Allah, The Almighty, Says (What means): "O ye who believe! Save yourselves and your families from a Fire whose fuel is Men and Stones." [Al-Tahreem/6]. And Allah Knows Best.
What is the ruling on working for a company that manufactures cigarettes?
All Perfect Praise is due to Allah, The Lord of The Worlds
Working for a company that manufactures cigarettes is forbidden because smoking itself is forbidden, so its production, selling and circulation are forbidden as a result since they aid in committing sin, and whoever aids in that is like the sinner himself. And Allah Knows Best.