Must be a bug.

Sorry for the delay. Hopefully the below info will get you going with the Open Graph stuff.

First, please note that Facebook has changed how they treat the 'read' action. Previously, it was given a lot of weight and displayed prominently in a user's Timeline. Over the last few months, they've drastically reduced the chances that it will be displayed prominently. The action can still show in that friends 'news ticker' of friend activity as well as activity on your site, but it will not show up in the user's Timeline as a prominent entry.

With that out of the way:

What is the "Object Type" if I wanted to post an article that the user read? There is no such thing in open graph help guide.

The Object type should be 'article' and you should check the "Built in FB object" box. You won't need to create anything in the Facebook Open Graph developer area as 'article' is a standard object type.

How should I set up a simple read and article action in Facebook?

After you've create the Object above to show on some of your content, create a Custom Action. The action type is 'news.reads' and again is a Built-in FB Action. Associate that with the object you created, set it to trigger only once and automatically after 10 seconds of the user visiting a page.

"Error: The action you're trying to publish is invalid because it does not specify any reference objects. At least one of the following properties must be specified: website."

This means your object is not setup properly or something else is wrong with the page. The og:type of 'website' should only be set on your home page. If you're seeing this on other pages, that's a problem. Let us know a URL to test on, and we can gladly look into it.

"Error: Application with ID 306047639503691 has not been granted the capability to use the property fb:explicitly_shared."

You won't be able to use the explicitly_shared capability until your action has been submitted and approved by Facebook. Make sure you're using a test user or the Facebook administrator account to test with.

If the user skips this step the user still can register on the website which is not a good thing for me. Is there any chance to build in the new version that if the user skip giving access to the open graph then the registration process will cancel as well?

No, that's not allowed by Facebook. They require that you have an app that degrades gracefully. If a user doesn't want to give permission to post to their wall, you shouldn't deny them access to your site, just deny them access to the features that they don't want. There's no way to require a specific permission other than the basics (email and profile information).

2. Can I store in any way in the mysql database if the user shared, tweeted the website? I know there is a respond from facebook, twitter but we have to modify the JFBconnect file to get this response what we don't want. We need this because we want to assign the user to a trial membership (akeeba subscription) if they click on facebook share or tweet the website but now we can't get these data unless we modify components/com_jfbconnect/controllers/request.php file.

JFBConnect doesn't currently have a way to track that. A feature for tracking that activity is being planned. With the next release of JFBConnect, we'll be adding an easy way to an admin to post any URL of your site to a Facebook Page or Group, Twitter Stream or LinkedIn Company page. The release after that should have tracking abilities and, we're planning, a way for user's to be able to post content to their own FB Page, Twitter Stream, LinkedIn profile, etc. The tracking will be for all of this activity and we're likely going to implement the same type of tracking for leaving a comment or clicking a Like.

One thing to note is that we already do track Open Graph Actions. So, if you setup a custom Open Graph action of "I want" or "Think this looks interesting" and put it on a page, when a user clicks that button, enters a message (and optionally tags a friend or place), that action will be stored in the database. That may be what you're looking for already, but if not, let us know.

Thanks,
Alex