How to remove an activity from a case or file an activity on several cases
You USED to be able to do this but the system was changed a little over a year ago so that it now removes the original from the non-case side - see Items not showing up on activity log.

There is the copy-to-case action for most activities on a case but it's not available for Emails/Inbound Emails. The reason might be that historically the "spirit" of CiviCase was to prevent something that might be challenged legally, i.e. if someone sends you an email it shouldn't be altered, to preserve the record, and that got applied broadly since internally it's similar code to other actions. You could try temporarily removing this line in CRM/Case/BAO/Case.php and see what breaks: https://github.com/civicrm/civicrm-core/blob/5.18.3/CRM/Case/BAO/Case.php#L2621. You might also need to grant the "edit inbound email basic information" permission in your CMS's permissions to see the action link if you aren't the admin user, but I don't think it's needed.

If you're comfortable with editing the database directly, find the internal activity id (hover over the view link on manage case and it's the number after the aid=), and then you can do DELETE FROM civicrm_case_activity WHERE case_id=1 AND activity_id=2, where you replace 1 and 2 with the actual case_id and activity_id. That should unfile it.