If you are a VIP member, right under the "Welcome, looser43!", just under the main menu at the top of the page, you will get a message informing you that you have new mail.  I will send you a test PM after I post this...did you see the alert?

Note that you cannot receive PM's unless you are VIP.  If you are expecting mail, and not getting the alert, check your membership status...it may have reverted to basic membership due to non-payment of the fee. This can also happen if there are changes to your CC info, or if the CC you are using has recently been renewed.

You should not assume that the person you are PM'ing has VIP status. They may not be getting your PM's.

is the "unread" mail that was sent be related to whether or not the person I sent the mail to has read it or not, rather than whether I have re-read my own sent mail.

I receive an email from TER that I have a PM; do not recall now if that was a selectable option.

It's found in "Edit Profile".  The default setting is to have TER  notify you via email when you have a PM, but you can turn it off.

If a person does not have VIP status, is it possible to send them a PM? I would think that the PM feature would not be selectable if the recipient doesn't have VIP status. Or at the very least, the PM should bounce back to the sender. Am I correct?

PM's can be sent to any TER ID (identified by the envelope that appears next to the name when posting).  

You can only see your PM's when you are a VIP.  There is no way to know who is a VIP, and the is no bounce back when you send a PM to someone who isn't.

It is one of the flaws in the system.