Maged,
Thanks for the input. That helps a lot to narrow things down. Doesn't sound like it's IPv6.
I've been looking more at your SSL configuration. You're using the SSL library called "NSS" for cURL vs what we use (and I think is more common), OpenSSL. It seems like the connection to Facebook simply can't be negotiated at all using the SSL protocol. I found a similar issue in a Stack Exchange post below:
stackoverflow.com/questions/7179216/php-...t-path-access-rights
I'm not sure how much control you have over your server, but if it's possible to try and rename that file (which I believe will clear the SSL cache), that may help. You also mention that your site from another server. It's possible that the old NSS database, either from your old site, or whatever is on the server now, is simply corrupted or holding incorrect (stale) data for Facebook.
So, with that, I don't think I've really given you an answer, but hopefully something you, or your server host, can look into. I really don't know if this is the issue, but it seems like some sort of failure to communicate over SSL. Our Disable SSL Validation setting still requires SSL communication, we just disable some checks to verify how secure the connection really is. You're site doesn't seem to be getting that far.
Keep us posted on any of the above or anything else you might come up with!
Alex