The only place that I know of to find the information that you are looking for would be in the AUDIT_LOG table of the project.
This table does not record the insert, update, or deletion of folders but does record them for the entities (ex. Tests) in the folders. Since the entities were moved to the ''Recycle Bin'' you should be able to find the tests that were moved and get the Test ID of the test. Using that information, you can query the AUDIT_LOG table to find out who moved them. This query can be run from within the Site Admin.
SELECT * FROM AUDIT_LOG
WHERE AU_ENTITY_TYPE = 'TEST' AND AU_ENTITY_ID = ###
ORDER BY AU_TIME_DESC