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").
<customUI xmlns="http://schemas.microsoft.com/office/2009/07/customui">
<backstage>
<tab idMso="TabShare">
<firstColumn>
<taskFormGroup idMso="GroupShare">
<category idMso="Share">
<task id="SayHelloTask" insertBeforeMso="ShareWithPeople" label="Say Hello Task">
<group id="HelloGroup" label="Say Hello" helperText="This group contains say hello button.">
<primaryItem>
<button id="SayHelloButton" label="Say Hello" onAction="ThisDocument.SayHello" />
</primaryItem>
</group>
</task>
</category>
</taskFormGroup>
</firstColumn>
</tab>
</backstage>
</customUI>
Download sample document (Doc1.docm)
The full list of built-in "idMso" names can be downloaded from Microsoft