I don't see many Jews with these characteristics either. The only descendents of my grandmother are my father and me. My father has a "Khazar" look too (less than my grandmother though). Not that much when he was young, but now he does. I don't, but I don't look like anyone in my family and new people at my synagogue often think I'm a convert. At my university people thought I was Polish or German. O_o lol
I don't know much about Khazar names either. But according to the family tradition Handkan means "the hand of the khan" (Khazar emperor). We're supposed to descend from the khans. It's totally normal you haven't heard of that name. It's extremely rare and strangely there are very, very few records on http://www.jewishgen.org/jri-pl/http://www.jewishgen.org/jri-pl/ which makes me think the family was in Russia before coming to Poland, but doesn't explain why I just can't find my great grandfather on it or something... Handkan is a surname (like Smith), not a first name (like John). I can't tell you how it is written in Hebrew, because my father never let me go to the cemetery with him.