As far as I know, while using Excel Import, one has to go to Data Import-> Import to check if there is any error.
Is it possible to detect that Excel Import has encountered an error and display some kind of popup message without leaving the entity overview page?
No, CRM does not allow you to extend the data import functionality. CRM data imports are parsed and handled by the asynchronous service, so what you are trying to achieve is not technically possible even if CRM did allow you to extend the import functionality.