How to set the path for the public directory?
You can maybe create a symlink of /var/www/sites/default/files to /var/www/drupal/sites/default/files. Assuming you're inside /var/www/drupal, the command to do that is:

ln -s ../sites/default/files sites/default/files


So with this in the File System configuration page, you can just use the default path "sites/default/files".

But you can also just transfer /var/www/sites/default/files into it's new and final location though just to make things cleaner and organized, if you want.

Updated

Since you have noted also that "../sites/default/files" almost worked maybe you just need to update the permissions of the folder to maybe 775 and try to clear caches.

Hope everything works for you.