What you could do is have the results of the copy be sent to a log file in a particular location so that you could have a shortcut on your desktop to that location and look at the logs.
To do that, your .bat files would direct the output of the command like this: copy *.* > c:\results.log
In the log, you will see the files that got copied and at the end it will say x number of files copied. If it says that, it worked. Also, you might be able to find a program from the web that allows you to send email from DOS and attached the log file.