Then this nutcase is a Jew but not Jewish?
Incorrect, sorry if I was unclear. Every Jew is Jewish. Everyone who is Jewish is a Jew. Just like everyone who is Polsih is a Pole and every Pole is Polish.
Judaism is a religion that if you are born into it (or convert to it), you can never become "not Jewish". You can only decide to not behave in accordance with the religion. Hence the millions of non-observant Jews in the world. They are Jews but either are ignorant as to what Judaism entails (this is the case for the vast majority of non-observant Jews), or they actively decide not to follow the religion. But they are still Jews.
So if Gadaffi was Jewish, that means he and I share common Jewish ancestors and are both charged with keeping the Torah.