Is the money taken by the employee in exchange for completing a transaction for his cousin in a regular manner, and with the help of his colleagues, permissible or forbidden?

May Allah reward you for your steadfastness in your religion and your fear of the unlawful. What you described regarding the transaction with your relative appears outwardly permissible, because your colleague completed it according to procedures. However, caution is necessary, as people may call things by names other than their true reality, such as bribery being called "halawa" (sweetener) or "ikramiyya" (gratuity), and usury being called "fa'ida" (interest).