Hello @Roel Evers
Thanks for sharing the Scenario on this forum!
Regarding your first issue, it's possible that the copy activity and pipeline are not being discovered by Purview because they are not considered assets in the same way that tables are. Create resource set pattern rules
Regarding your second issue, it's possible that the file is not being recognized as a "File" asset because it is part of a resource set.
Customizing resource set grouping using pattern rules
Regarding your third issue, it's possible that the asset name is being cached by Purview and needs to be refreshed. You can try clearing the cache.