The goal is to have a user paste a URL in textarea, and it outputs not a simple text or URL but displays that image from the URL?
http://www.thecampussocialite.com/wp-content/uploads/South-park-VS.-Family-Guy.jpg
Can this be done? Are there any examples?
The textarea is simply not capable of that.
You need to use a rich text editing control, instead of the basic HTML Text Area tag. Popular options are: