I have tried learning more about the Mail Envelope Index, but I still have not found a (simple) straight forward answer to what that is, and when & why one would use it (and, are there any pitfalls in doing so?).

This should help
http://jimmitchell.org/2008/02/20/rebuilding-the-mail-envelope-index/
The Author is the creator of Yasu
Mark

Many thanks, Mark, for the link & enlightenment.

I don't know that I need this, but when/if I do, well, I'll be ready.

Normally, I use Cocktail for housekeeping such as this, but, AFAIK, this particular feature is not in Cocktail.

Glad to help. I use Onyx and Cocktail also and never knew what the Mail Envelope Index in Onyx did either, though I would run it once in a while. Strangely the site I sent you was the first one that came up in Google.

I wonder how this is different from rebuilding the Mailbox using Mailbox > Rebuild from the Mail menu?

I am getting all my info from the internet. Below is the site for the macworld article, but it is almost 5 years old. I am using 10.6.8 and i CAN REBUILD MULTIPLE MAIL BOXES FROM THE MENU COMMAND (the article says you can’t) . According to the article below they are the same.
I also just went into the mail folder in users as per the Jim Mitchell article and that works fine. I have only 350 emails and it rebuilt very quickly. I don’t have the personal expertise to say if they are all comparable, but they seem to be.

http://www.macworld.com/article/56673/2007/03/mailfix.html
(excerpt from the Article) A possible fix for a slow Mail app
by Rob Griffiths, Macworld.com   Mar 8, 2007 1:00 am
You can do this from within Mail using the Mailbox -> Rebuild command, but you’ll have to do so one mailbox at a time—the menu option is grayed out if you have more than one selected.