Lets say we delete various files , from hard disk A, with shift - delete , windows 8 . Then we copy new files to the same hard disk A. Then those new files we just copied there to hard disk A, we copy to a USB stick , or to a DVD. Those new copied files are now laying to a total different hard disk drive B . Is there anyway , those new copied files that came from the hard disk A , but now laying in the different hard disk B , to include any traces of the previous deleted files from hard disk A , so that recovery software be able to recover , if we run the recovery software from the hard disk B ? I only used normal windows explorer copy and paste, for copying the files.
It depends on how you copy them.
If you do a bitwise copy, aka exactly replicating every single bit from one disk to the other, you are copying over "slack space" which is where the recovery tool will work.
If you do a file copy, then the empty space of the disk is ignored, and only the segments of data attached to a file header will copy over. The recovery tool cannot see the slack space attached to the other disk.
TL;DR; It depends how you moved it.
If you have never moved the files to Hard Drive B, it won't carry over the traces.
The way the files are stored on a hard drive would leave traces of the file on Hard Drive A (gets overwritten over a period of time). The difference is that when you copy files from one hard drive to another, it only copies the files you selected and skips the rest of the platter (skips the traces of deleted files).
Now, there's an exception to this. If you Clone the hard drive (I.E. Sector-to-Sector copy), that can carry over those traces because cloning the drive itself brings over all data on the platters (deleted or not).
Hardware cloning is usually a sector-to-sector copy, moving over everything to be the exact same (including partition sizes).
Software cloning is more flexible, because it can do both an exact copy of the platter (bringing over traces of the deleted file) or copy only the existing files (skips deleted files). Don't misunderstand when I say deleted files, I do not mean files left in the Recycle Bin.
I could be missing some information here, but I hope this answers your question.