Facebook – Are Facebook accounts always re-attainable

In fact mobile setting are only for your Facebook account (to receive alerts by SMS for example). Using this page you can't either confirm a cell number from China for example.

This is the URL you should use: http://www.facebook.com/confirmphone.php (this page will offer the complete list of countries).

Alternatively you can also use this page: https://www.facebook.com/update_security_info.php (Singapore is available in the list of countries), this trick was found on the link provided by @marko4o (http://bugs.developers.facebook.net/show_bug.cgi?id=16375#c126).

The i.p address which you have been provided is class A i.p address i.e it is meant only reserved for private use and can be used inside any organization(i.e it might be from any of the shared computers of the network). Facebook wont try to break into your account, click on i didn't recongnize and navigate to next tabs, usually facebook would trace the ip location too ,if you suspect it's an issue then report here