Auto Like and Send in every article

To specify the image you want used when the Like is clicked, you need to use Open Graph tags. Open Graph integration in JFBConnect is enabled by default, but you can check the setting in the Social Configuration -> Open Graph tab. Then, all you need to do is add the following tag to your article to specify the image you want used:
{JFBCGraph image=http://site.com/link-to-image.jpg}
It will take Facebook 24-48 hours to update the image for new like's, so if you try right away, you may not see a difference. You can use the debug tool to see what Facebook 'sees' on your page from the Open Graph tags:
developers.facebook.com/tools/debug

Finally, you can also use the Open Graph tab in the social configuration area to specify a default image, in case you haven't explicitly set one on a page using the tag above. This is good for using your logo, like you mentioned, just to make sure some random image isn't shown. To do so, add the following tag into the defaults box:
image=http://site.com/good-default-image.jpg

Hope that helps,
Alex

P.S. There's also a setting in the Open Graph tab (I think that's where it is) to have JFBConnect try to automatically detect the first image in a Joomla article or K2 item and use that.

Let us know how that goes!