Did you create the directories within /var/www/app_name, or were they created by capistrano?
Regardless, the issue you have is that /var/www/app_name/current should not be a directory - it should be a symlink to the current release within /var/www/app_name/releases/. The failure is caused when capistrano has finished creating the new release folder within /var/www/app_name/releases/, and is trying to symlink /var/www/app_name/current to it.
You might be able to fix your issues by renaming /var/www/app_name/current (so you have a backup if things go wrong), and creating a symlink from /var/www/app_name/current to the most recent release within /var/www/app_name/releases/, and then doing a cap deploy. (Delete your backup of current if this works).
As far as best practice goes whatever you do, do not use root. Instead, set up a user (or use an existing user) that has only permissions to the required directories (didn't read your scripts closely, but probably just /var/www/app_name.