Finally got it figured out after a lot of struggle.
Using Cygwin I was able to get a copy of unzip, which I put in the Program Files (x86)\Drush\GnuWin32\bin directory (which is in my PATH environment variable). This cleared up the errors listed about unzip.
As for the mime-type issue, I had to manually edit includes/drush.inc to add '.js' => 'application/javascript'
to the $extension_mimetype array. See http://drupal.org/node/1825946 for the discussion that led me to this.