How to set up a repeated “field set” with unlimited time and event?
This is possible using the "double field" module (https://www.drupal.org/project/double_field) and using the "field permissions" module (https://www.drupal.org/project/field_permissions). 

In the users' account settings add a double field, set it up as date/text combination, allow unlimited values. Use the field permissions setting to permit access to specific roles only.

This does't allow to use images to be used, unfortunately.

An alternative idea is using a paragraphs field (https://www.drupal.org/project/paragraphs), where you can define a paragraphs item with any fields you need. Again, set to unlimited allowed values and setting field permissions.