Is it possible to extract the search criteria used to save the search results to a folder when exporting the contents of that folder?
No, the query is not visible to an export operation. The contents of a folder being exported is not always a true representation of what was originally saved into that folder. Contents of the folder may have been altered by a move, copy or a remove operation. The query that was used to save the search results to a folder can be obtained by running an audit search or reviewing the 'Case Searches' page.