File

You can use file() to grab the html of other webpages, but is there a way of specifying the headers that the php sends to the grabbed page ? For example if I want to specify Cookie: etc.

For this you will need a socket connection:
http://www.php.net/manual/en/function.fsockopen.php

also take a look at the snoopy class (http://snoopy.sourceforge.net) which does a lot of the http work for you..