Is it permissible to invest money with a merchant through an installment plan – via a friend acting as an agent – and must the merchant know that I am the owner of the money?

The validity of the transaction between your merchant friend and your agent friend, who acts on your behalf in investing your money, is sufficient if your merchant friend is aware of it. However, your friend guaranteeing your money is impermissible, because an agent is a trustee who does not guarantee anything unless there is transgression or negligence. Agency is a contract of facilitation and assistance, and guarantee contradicts that. If the agent transgresses, then he becomes liable.