Facebook has Developer Plugins that do exactly what you want.
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To answer your second question, your goal is to have organic traffic comming to both your site and your Facebook page and you want to encourage cross over. When I was working on my author site a while back before I moved to blogger to reduce maintenance, we had a photo roll with me at a book signing in a box in the side bar and a like button there, for the page (we used the page badge for that).
Then, at the bottom of each blog post we had another like button, clearly segregated by theme (and the style of the button since the page button was the badge) which was for the post. This like button would be incorporated into the various share links, but typically you see the "Like" and "+1" buttons broken out from the other buttons for things like "Reddit" and what have you.
So in summation, you'll want both. It's important to understand that the Like button for URLs will not (most likely) work as you want for your FB Page (as in, copy/pasting the URL from your FB page into the code above in an attempt to have people
Like your Facebook Page). As I understand it, the users would be Liking the URL (of your Facebook page) as opposed to the Page itself. You should use the Page Badge for this function, and the Like Button I pulled out of the FB Developer Guide for posts and actual Web pages on your site. (You could also use a Like Box in place of the Page Badge if you like.)
You can find the page badges here: http://www.facebook.com/badges/ (It's behind a login or I'd paste code too).
I hope I've answered your question. There's a lot to do with the Developer Plugins from Facebook, when I maintained my author blog on my own servers I used the plugins for comments as well as likes, which is especially useful because each comment causes a News Feed entry to happen on the Commentor's Friends' Feeds (as well as their own. ie: "Bob Barker commented on BLahBlahBLah..."). I'm getting off topic now.