Word 2007 Adding Mysterious Line on Save to Images

That sounds like you unknowingly made a change to the header or footer section of the document.  See this link for things that can be placed in the header/footer, or how you can clear the header/footers.

        http://office.microsoft.com/en-us/word-help/insert-headers-and-footers-HP001226486.aspx

No, there is nothing whatsoever in the header or footer.

I know you've said it is not a border, but try following the removal procedures in this link:
                    http://office.microsoft.com/en-us/word-help/remove-a-border-HP005189263.aspx

make sure the line is not part of a custom border or style sheet, which requires different procedures
                     see    http://sbarnhill.mvps.org/WordFAQs/BorderArt.htm

The link for Office doesn't completely apply because it's Word 2007 not 2003, but looking at the Picture Tools and Format Tab on the ribbon, there is no line chosen under Picture Border, and nothing chosen under Effects.

I don't see where on the 2nd link it provides instructions on removing a border, but I am experencing this on multiple documents, all of which I've created from scratch.

It was the compression settings.  It was set to "Automatically perform basic compression on save"  I turned that off and problem fixed.  Thank you!