Stars next to name
The yellow star indicates that they have paid for the Gold level of premium membership; you may also see a crown indicating Platinum membership and a diamond indicated Diamond membership - see https://www.chess.com/membership
Your display name is your user account name, and you can change it once yourself - see https://support.chess.com/customer/en/portal/articles/1444791-how-can-i-change-my-username-. If you want to change it again after that one time, you will need to contact the Support team via Help --> Ask a Question --> Contact