Is there a way to detect if the Facebook Javascript SDK loaded successfully?

You should load the Javascript Library Asynchronously and put all your FB related functions inside the window.fbAsyncInit method:

<div id="fb-root"></div>
<script>
  window.fbAsyncInit = function() {
    FB.init({
      appId      : 'YOUR_APP_ID', // App ID
      channelUrl : '//WWW.YOUR_DOMAIN.COM/channel.html', // Channel File
      status     : true, // check login status
      cookie     : true, // enable cookies to allow the server to access the session
      xfbml      : true  // parse XFBML
    });

    // Additional initialization code here
  };

  // Load the SDK Asynchronously
  (function(d){
     var js, id = 'facebook-jssdk', ref = d.getElementsByTagName('script')[0];
     if (d.getElementById(id)) {return;}
     js = d.createElement('script'); js.id = id; js.async = true;
     js.src = "https://connect.facebook.net/en_US/all.js";
     ref.parentNode.insertBefore(js, ref);
   }(document));
</script>

This code loads the SDK asynchronously so it does not block loading
other elements of your page. This is particularly important to ensure
fast page loads for users and SEO robots.

The URLs in the above code are protocol relative. This lets the
browser to load the SDK over the same protocol (HTTP or HTTPS) as the
containing page, which will prevent “Insecure Content” warnings.

The function assigned to window.fbAsyncInit is run as soon as the SDK
is loaded. Any code that you want to run after the SDK is loaded
should be placed within this function and after the call to FB.init.
For example, this is where you would test the logged in status of the
user or subscribe to any Facebook events in which your application is
interested.

A quick example is the following:

<div id="fb-root"></div>
<script>
  var isLoaded = false;
  window.fbAsyncInit = function() {
    FB.init({
      appId      : 'YOUR_APP_ID', // App ID
      channelUrl : '//WWW.YOUR_DOMAIN.COM/channel.html', // Channel File
      status     : true, // check login status
      cookie     : true, // enable cookies to allow the server to access the session
      xfbml      : true  // parse XFBML
    });
    isLoaded = true;

    // Additional initialization code here
  };

  function checkIfLoaded() {
    if(isLoaded) console.log("LOADED!");
    else console.log("NOT YET!");

    return false;
  }

  // Load the SDK Asynchronously
  (function(d){
     var js, id = 'facebook-jssdk', ref = d.getElementsByTagName('script')[0];
     if (d.getElementById(id)) {return;}
     js = d.createElement('script'); js.id = id; js.async = true;
     js.src = "https://connect.facebook.net/en_US/all.js";
     ref.parentNode.insertBefore(js, ref);
   }(document));
</script>
<a href="#" onclick="checkIfLoaded();">Check</a>

(Just clicked the check link a couple of times)

---

Please note that you can still construct the Login Link server-side and WITHOUT JavaScript. Example using the PHP-SDK:

$loginUrl = $facebook->getLoginUrl();
...
...
<a href="https://stackoverflow.com/questions/5334977/<?php echo $loginUrl; ?>">
    <img src="http://static.ak.fbcdn.net/rsrc.php/zB6N8/hash/4li2k73z.gif">
</a>