Youtube – From which timestamp of a YouTube video is the preview thumbnail taken

You have to be a partner if you want the option of choosing what custom thumbnail represents the videos you upload.

This was brought up in the Partner Forum of YouTube in Custom Thumbnail update:

As many of you may know, the custom thumbnail program has been running for some time now, allowing our partner community to select custom images for their videos.

And is mentioned also in Flagging Videos > Spam of Content Reporting and Removals

Misleading thumbnails refer to users who chose a thumbnail that is not representative of the actual video content in order to gain views. Only those in the partner programme are allowed to choose their own thumbnails.

You must be doing something wrong, here's the right way to share YouTube videos on Facebook wall, go to that video page on YouTube that you want to share, then click on Share button, you'll see a Facebook icon there, click on that icon, a small pop up window will open, Facebook will ask your login details, if you are already logged in Facebook then you'll see your video embedded in a post, just give a title to your post and then you'll see your video posted on your wall rather than links :)