Hey there,
Just wondering how I can leave my Facebook app connected to a user as soon as they launch the app and not click on LOGIN now and then.

Hy buddy.
At first you can't access your app without being logged in. In fact you need a so called access token for your app.
To get more informations how to generate the token look here:
https://developers.facebook.com/docs/opengraph/using-app-tokens/

And here is a code example from my events_facebook import to get events from specific page and display it in silverstripe

<?php
require_once 'fb_api/facebook.php';

// Authenticate
$facebook = new Facebook(array(
   'appId' => 'your_app_id',
   'secret' => 'your_app_secret',
   'cookie' => true, // enable optional cookie support
));

// Attempt to fetch SpiritGallery's events page
// place access token for your fanpage after events?access_token=
// also place your fanpage that you would like to pull events from in $events=$facebook->api('/fanpage_here/
try{
   $events=$facebook->api('/radionationband/events?access_token=copy_your_token_here');
}catch (FacebookApiException $e){
   error_log($e);
}

// Iterate through each event
foreach ($events["data"] as $event){

I hope this helps for the first time.

Does this code make/or cover the entire website having the user logged in?
Thank you for your response it's much appreciated =)