Not Working

I. UPDATE to latest OS X (10.8.3) and SAFARI (6.03) browser that was launched June 20, 2013. It addressed exactly the Facebook issues you are experiencing.
http://www.macobserver.com/tmo/article/apple-improves-facebook-performance-in-safari-6.0.3171

II. UPDATE JAVA

III. ENABLE COOKIES.
Some customers have reported problems accessing the website since the latest update issued for Safari on iOS devices (iPhone/iPad). It appears that this update by Apple has turned cookies off by default, and our website (like many others) needs cookies enabled to make it work.

To enable cookies in iOS Safari:

1.Go to the Home screen by pressing the Home button or by unlocking your device.
2. Select the Settings icon.
3. Select Safari from the Settings menu.
4. Select Accept Cookies from the Safari menu.
5.Select From visited from the
Accept Cookies menu.
6.Press the Home button to return the home screen.
7.Select the Safari icon to return to Safari.
8.Before the cookie settings change will take effect, Safari must restart. To restart Safari: Press Home twice to bring up open apps, find Safari and press and hold the icon, then select the red symbol near the top left of the icon.
9.Select the Safari icon to return to Safari.

IV: DISABLE ADD-ONs, especially some of SAFARI's privacy-protection extensions like FACEBOOK DISCONNECT:

When I first started, I had problems connecting CTS via iPad/iPhone to my Safari Browser with Facebook Connect. After a tip and some tests, I found out the culprits were Safari's privacy/data protection extensions (Facebook Disconnect, DoNotTrackMe, Ghostery, etc,...). So I deleted them, and I was able to connect to my friends on CTS through Facebook.
Problem is, some of them still wished to use these privacy programs, so here's a set of instructions to allow to use them, and still connect to CTS Facebook friends:

(A) Safari Extensions that will NOT block Facebook Connect (no deletion/settings needed):

1.WOT
2.AdBlock
3.JavaScript Blocker
4.Dr.Web Link Checker
5.Bitdefender TrafficLight

(B) Safari Extensions that WILL block Facebook Connect no matter what (remove):
1.Facebook Disconnect.

(C) Safari Extensions that need to be configured to allow Facebook Connect to work:
1.DoNotTrackMe.
After installation, you will see a new button on the toolbar with a crosshairs symbol. Click to open window, click on the gear button top left, scroll down on list, and slide to red the following: Facebook Beacon, Facebook connect. Close.
2.AVG Do Not Track.
After installation, you will see a new button on the toolbar with a eye symbol. Click to open page, scroll down on list, and uncheck the following: Facebook connect, Facebook Social Plugins. Close.
3.Ghostery.
After installation, you will see a new button on the toolbar with a ghost symbol. Click to open page, scroll down on list to Widgets, expand list and uncheck the following: Facebook connect, Facebook Social Plugins, Facebook Social Graph. Close.

(D) Special Safari Extensions:

1.Incognito:
It will not break Facebook Connect, but it will NOT allow you to connect to this forum via the Facebook login button. Solution: On the Safari Menubar, navigate to Safari, Preferences, Extensions, scroll down to the Incognito box, uncheck the "Facebook" button, restart Safari.
2.MyPermissions.
This extension reports what Facebook apps are allowed what to access, privacy/permission wise. You can click and add the CTS Facebook app to the Trusted pile, but you have to pay 4.99$ a month to do so. No really a necessary app.

And those are all Extensions I had to deal with, or configure for other friends.
CAVEAT: This deals with CTS and Facebook Connect issues. They may cause other problems with other sites or apps (none that I can see, though).