Mail Application-I'm receiving failure notices on emails which I did not send. ???

As you guessed, someone else is using your email address. There is nothing at all you can do about this unfortunately.

This could be due to someone faking your address as the sender when sending spam. In this case, there's nothing to be done.

However, it could also be due to someone hacking your e-mail account. The fact that you don't see the messages in your Sent mailbox does not necessarily mean anything, since hackers have been known to delete any messages they sent from the Sent mailbox. You need to immediately change your e-mail password to be sure the account is secure.

In addition, some e-mail accounts provide features that make it easy for hackers to create a "back door" to get back in even after you change the password. Case in point, GMail's account delegation. Check with your e-mail provider and make sure no such feature is being abused by hackers.

Just to keep you apprised, I changed the security password to my email account, and am pleased to report that after a day, failure notices have ceased.

That's an indication that your account truly was compromised.

Your Mac has nothing to do with your account being compromised, once that happens it will have exactly the same outcome on a Windows machine.

Your Mac was not compromised, your account was.