The proper way to wear a yarmulke is to wear it on the Sabbath, Shabbat, or when in Shul. One should wear a yarmulke whenever everyone else is wearing one.
Yes
Orthodox Jewish men wear the Yarmulke all the time. Other Jews (male and female) decide for themselves whether or not to wear it.
No. You don't have to be orthodox to wear a yarmulke. In fact, you don't have to be Jewish. Some modern liberal Jewish congregations have pronounced the yarmulke optional during services or have dropped it completely, but it is still primarily a matter of individual choice. There are even some Jews who are quite secular, do not regularly pray or attend synagogue services, but who wear the yarmulke in general daily life.
Yarmulke
Yarmulke and Yashmak
Orthodox Jewish men wear a yarmulke (a skullcap). It begins with the letter Y.
Jewish teachers (rabbis) typically wear a kippah or yarmulke on their head.
Yarmulke (Yiddish) or kippa (Hebrew).
A kippah, or Yarmulke, or a hat or streimel.
A kippah, or yarmulke. http://en.wikipedia.org/wiki/Kippah
Justin Bieber's manager, Scooter Braun, is Jewish. In order to attend his manager's wedding, Bieber had to wear a yarmulke like all of the other men in reception. It does not signify anything other than his respect for his friend and co-worker.
The Zuchetto has eight panels, while the Yarmulke has many different shapes, sizes, and patterns. And they have some difference in purpose. The original purpose of the zuchetto was to cover the tonsure, most likely for warmth in the days before central heating. It has become a symbol of tonsure, and therefore the zuchetto is proper for wear by all clerics, not just priests and prelates. The yarmulke (actually called a kippa (kee-PAH) by the majority of Jews) is a symbolic item. It is meant to be a visible acknowledgment that HaShem (The Creator) is always above us.