How can I perform copy/move file operations while ensuring data integrity?

Best tool for this job is rsync. It can quickly sync deep directory trees and ensure that data is transmitted safely.

Many NAS boxes support rsync protocol natively. If your NAS does, you're in luck.

If you use Linux as your client, most likely you already have rsync installed. To copy content from NAS to your computer, use this command line:

rsync -av user@host:/some/dir/ /some/local/dir/

If you want to 100% ensure that data was copied safely, add --checksum option - but it will make copy a bit slower.

On Windows, you could install grsync - it has nice GUI, but it also installs command line rsync Windows executable, which you could use with exact same command line as used above for Linux.

Also, you can copy data back to NAS using this syntax (swap source and target):

rsync -av /some/local/dir/ user@host:/some/dir/

If you want something with a GUI, then Teracopy is where you want to go, taking advantage of the "Always test after copy" option.

If you're happy on the command line then Robocopy is where I'd start (and whilst you're there, look up the "Verify" command).