Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.
What is the ruling on a man leading his sisters in obligatory prayer, or Taraweeh prayer (Night prayer in Ramadan)?
The prayer of the man who leads his wife, or sisters in prayer is valid, but it is more rewarding for him to offer obligatory prayer in the mosque. And Allah Knows Best.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.
I`m a cashier at a branch of the Civil Service Consumer Corporation. Sometimes, customers pay more than the due amount. Do I have the right to take that extra money knowing that I cover shortages from my own salary?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
If customers give that money willingly to help you with the potential loss then this is permissible. And Allah The Almighty Knows Best.