Hey @jasminchang,
The Events Calendar does not have any of its own “Export” features, so if you are referring to a specific “CSV Exporter” tool on your site, it is not something we make and so I can’t say whether it will transfer URLs or images, etc.
If you’re using the WordPress “Export” tool, which exports XML files, then YES, the association of an event to its featured images will be included in the export file information, though you will need to transfer the images of your media library over to your new WordPress site. There are many resources online for learning to do that, so if you’re not familiar with this process, definitely look into those.
As for having the “same URL”, this depends a bit on what you mean – if your permalinks settings are exactly the same, then the event title and the event date in the URL should indeed be the same.
Thanks!
George