Restoring a deleted or moved file is related to permissions a user has. Now if you are an admin, permissions won't matter. However, if you are a power user or standard user then performing this task will require permissions.
All deleted files land up in the Trash on the UI. An administrator can always browse the Trash and restore the deleted files. As far as the moved files are concerned, restoring them is possible if you know where the files has been moved. If not, then the administrator has to run a file audit report (has to be a subscribed feature on your domain) and learn about the source and destination folder for the file move.
UI operation to copy/move a file is very simple. Browse to the folder => Select the file => Select the copy/move action from the sub-menu => Provide the Destination folder.
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