One reason it seems unclear is due to the whole stereotype that Jewish people have a particular shaped nose, how could that possibly have any truth to it unless they were a separate race. I've always consider stereotypes to have a lot of truth behind them, otherwise they surely wouldn't exist in the first place.

Well Judaism is an inherited thing. Judaism is traditionally passed from a mother to her child. It's to do with the fact that the main aspect of Judaism is the covenant the Jewish people have with God by descending from Abraham. A person can be converted to the Jewish faith but it's a well complicated process. But compliant won't marry non-Jewish people, hence the fact they are steriotyped with certain features etc. In the past they would all be related! Some info:

According to traditional Jewish Law, a Jew is anyone born of a Jewish mother or converted to Judaism in accord with Jewish Law. American Reform Judaism and British Liberal Judaism accept the child of one Jewish parent (father or mother) as Jewish if the parents raise the child with a Jewish identity. All mainstream forms of Judaism today are open to sincere converts. The conversion process is evaluated by an authority, and the convert is examined on his sincerity and knowledge.

Traditional Judaism maintains that a Jew, whether by birth or conversion, is a Jew forever. Thus a Jew who claims to be an atheist or converts to another religion is still considered by traditional Judaism to be Jewish. However, the Reform movement maintains that a Jew who has converted to another religion is no longer a Jew, and the Israeli Government has also taken that stance after Supreme Court cases and statutes.

It's called a skullcap (or variously called a Kippa, Yarmulka or Cappel by Jews - in England, you'd mostly hear Cappel). And, no, I don't wear one. The reason it is worn is to constantly remind Jews of God. In fact ultra-orthodox jews won't walk more then a couple of metres without one on their heads.