Ok, here's my understanding of the scenario. You have the cookie grabbing script on site A, and want it to be saved on the server hosting site B. If this is the case, you would have to have site A host some sort of script to forward all of the necessary information that the browser sends to the server at site A, and have a script setup on site B to write the information it recieves to a text file. If all you're interested in is the cookie, you could easily write a script to request the script at site B, sending the cookie. This would be accomplished somewhere along the lines of:
In fact, all the script at site A would need to do in this case is to recieve the cookie you want to steal via a GET or COOKIE variable ($_GET or $_COOKIE depending on how you want to do it) and use define the $cookie variable in the fopen function.
Now, the question of embedding it into a page is another story. If you were to actually gain access to a file, you could add the script at site A's code to a page on the target site and, using $_COOKIE, silently steal the cookie of every visitor to that site. You could also upload the script onto the target site in question and add the IFRAME html tag into any page there, and if you had a user view it, their browser would request the script, and send the applicable cookies.
Of course, you need to understand, you can't 'embed' a remote script into the site via IFRAME because browsers will only send cookies to the site domain the cookie is set to be sent to. Also, with the first option, if the PATH of a cookie is set, it will only send the cookie to a script in the PATH that the cookie is instructed to be sent to. This is why 'cookie stealing' is more complicated than writing a simple script (hence the name cookie grabber for my script).
As far as PHP functions you should be concerned with, start with...
and the global variables $_GET and $_COOKIE
I'm not sure I understand your comprehension of the $_COOKIE variable. By specifying no cookie name in the cookie variable, you will not return any value. The $_COOKIE variable is an array.