Wouldso would I redirect subdomain.domain.com to show targetdomain.com without changing the URL? To ensure that subdomain.domain.com/graph.php would really display targetdomain.com/graph.php?

I understand it is something related to mod_rewrite and .htaccess - I simply can't have it working!

You're close. You have to enable and employ Apache mod_proxy in your Apache config. Check this out link: http://httpd.apache.org/docs/2.2/mod/mod_proxy.html