The ThemeMin demo site at http://thememindemo.wordpress.com/ was using an older version of a Twitter widget that was deprecated due to API changes that Twitter made. We updated to a newer widget called the Twitter Timeline Widget, which is the one you’re using.
I put in a request to update the demo site to use the correct widget.
I looked for a way to remove the Twitter profile picture from the left of tweets displayed in the Layout Options inside the Twitter Timeline Widget, but I don’t see a way to remove them. Sorry about that!
Thanks for the note about the ThemeMin demo site using an outdated widget, we’ll get that fixed so it doesn’t cause anyone else the same trouble.