Can I use Google as a domain name registrar?

Google uses another company for registering domains.
They alllow you to sign up for Google Apps and register through their site and they turn around and register domains on your behalf through either GoDaddy or Enom.  They choose the company.

The nice thing about this is, the domain is automatically set up for use with google if you do this through Google.  You still have access to all the zone records through your Google Apps dashboard too.  It's very integrated.  

There is no folder system with FTP however and you will not be able to use anything but Google Sites for web pages with the default setup.  You would need to contact the Registrar being used to discover how to change name servers to set up externally hosted sites if you need one.