If you have no article in the frontpage buttons can't be showed.
About the shared image for Facebook it's ruled by the og:image meta tag.
Here is a quick checklist of what you can do with this plugin:
http://storejextensions.org/forum/view/19-fast-social-share/4069-re-wrong-image.html
and here is explained that it can be used as a module, so eventually showed in your homepage even if you have no articles
http://storejextensions.org/forum/view/19-fast-social-share/1603-re-using-fast-social-share-with-easy-blog.html
Regards
