Is it possible to detect that Reuters EIKON instance is not logged in and execute a login/sign-in in Reuters EIKON desktop instance from Excel VBA code via COM API and specify certain username and password programmatically (not via a pop-up window)?
No, this is not possible. It's possible to login silently by checking "Sign me in automatically" checkbox in the login splashscreen. It's also possibly to detect if Eikon Excel add-in is signed into Eikon platform and trigger sign in if it's not. See the following tutorial for more details.