How to recover deleted files from Iphone 4(iOS 4.3)
Deleted files? No. He's probably talking about things like Music Rescue, but they dont recover deleted files unfortunately, just take them off the phone without using iTunes. 

Your method of moving photos and movies is fine as you are only accessing the "camera" section of the iPhone's storage, and it is one of the methods that are supported.

I think while you moved files around your cache may have filled up or something happened to the copy paste while they were cut, unfortunately cut is a destroying action, and if you cant ctrl-z immediately, I would assume it's game over.