Facebook – Using a Facebook Page as a social plugin to get friend requests from website visitors

Register your domain to insights and see what you get. (Takes about a day to collect the data)

http://www.facebook.com/insights/

I wish I could vet this for you but the site I currently administer is currently down. I will update and test the moment I get it back up.

For Example, The Facebook Page though for the domain shows

• Likes
  • Daily New Likes
  • Daily Active Users
• Interactions
  • Daily Post Views
  • Daily Post Feedback

You can also just inspect the pages themselves in the case of the like button

https://graph.facebook.com/?ids=http://www.imdb.com/title/tt1375666/ This gives the object info for the Inception Movie from IMDB

{
   "http://www.imdb.com/title/tt1375666/": {
      "id": "110935752279118",
      "name": "Inception (2010)",
      "picture": "http://profile.ak.fbcdn.net/hprofile-ak-snc4/hs275.snc3/23312_110935752279118_3281_s.jpg",
      "link": "http://www.imdb.com/title/tt1375666/",
      "category": "Movie",
      "description": "Directed by Christopher Nolan.  With Leonardo DiCaprio, Joseph Gordon-Levitt, Ellen Page. In a world where technology exists to enter the human mind through dream invasion, a highly skilled thief is given a final chance at redemption which involves executing his toughest job till date, Inception. Visit IMDb for Photos, Showtimes, Cast, Crew, Reviews, Plot Summary, Comments, Discussions, Taglines, Trailers, Posters, Fan Sites",
      "fan_count": 22285
   }
}

The activity feed will take care of displaying stories both when users like content on your site and when users share content from your site back to Facebook. The Like Box is just an aggregation of Likes. Login Button and Friends Pile share the same display functionality.

I am not sure how the Live Stream operates have not had an event to test it with yet.

I don't think so, it is meant to be for your users to know about their friends. You could search for your site with site:facebook.com but not all profiles get crawled by Google.