Can I get just a simple computer name (without the domain name) from a fully qualified name (can be with or without a domain name)? Is it possible for a computer name to have a dot (.) sign in it?
No hostnames cannot contain a dot (reference Wikipedia and RFC 952 (see "ASSUMPTIONS") and RFC 1123). It is the delimiter between the hostname and the domainname. So you can simply do
string fullName = "foobar.domain";
string hostName = fullName.Substring(0, fullName.IndexOf('.'));
(With proper error checking of course, for the case that "fullName" is not actually a fullname).