Hi,
I have tested your plugin in the debugging section and it finds a video thumbnail but when I test the 'image downloading option' I get the error message below. This is also with both options (Save Thumbnails to Media Library & Automatically Set Featured Image) selected. Because of this issue the featured image isnt displayed and it should be.
✖ Error retrieving a thumbnail from the URL http://img.youtube.com/vi/dMH0bHeiRNg/0.jpg using wp_remote_get()
If opening that URL in your web browser returns anything else than an error page, the problem may be related to your web server and might be something your host administrator can solve.
Details: name lookup timed out
Can you help with this error at all?