Interestingly, the field of genetics recognizes "Jewish" ethnic genes. For example, the Askenazi Jews are the group whose ethnic origins can be traced to Eastern Europe. What's true is this genetic assignment is based on a high correlation between Ashkenazim and our associated long-standing social practice of Judaism as a religion. Certain diseases - like Tay-Sachs Disease, Crohn's Disease, Factor XI Deficiency, Familial Dysautonomia, to name a few - are far more likely to occur in Ashkenazi people because of how rare it was for them to inter-mix with other ethnic groups prior to about 1944 (they missed out on the would-be benefits of hybrid vigor). And it might not be all Jews, but it does seem like a disproportionate number of Jews do have curly - if not very curly - hair. Not sure why.
I'm all kinds of mixed, so honestly, the hair looks like a particular Jewish stereotype, but it could also come from elsewhere in my family tree ... just think it's more likely the Jew in me.
Well, sure. That's because the Ashenazim makes up roughly 80% of today's ethnic Jewish groups. As for ID-ing as Jewish, I can tell you as a New York Jew who's lived with and around European Jews most of my life, they don't hesitate to identify as "Jewish", quite proudly I might add

Those who live in Israel? I'd expect to define themselves as Jewish. They have a tremendous sense of Jewish nationalism, as a general rule, based on the socio-political and religious stake they have in that region of the world.