Is there a way to add a new service next to "Invite People", "Email", "Present Online", and "Publish as blog post"? I know you can add another source for the "save" and "open" tabs, but what about the "Share"?

Is there a way to add a new service next to "Invite People", "Email", "Present Online", and "Publish as blog post"? I know you can add another source for the "save" and "open" tabs, but what about the "Share"?

You can do that via "backstage" customization - i.e. you can embed your items in any of the built-in tabs in fact. You could do that either on document/template level (so that this new option is only available for a specific document/for a documents created out of specific template), or on "application" level by building an add-in for Word which adds this option.
Check out these articles:
Introduction to the Office 2010 Backstage View for Developers
Customizing Office 2013 backstage
Note that visual ribbon designer (in visual studio) does have support for the extending built-in tabs, you will need to extend office ribbon "backstage" by defining your "additions" in XML. For the "Share" tab you may need some xml like this (note the "idMso").
Download sample document (Doc1.docm)
The full list of built-in "idMso" names can be downloaded from Microsoft