In the 18th and 19th centuries, Austria and most German states made laws mandating that people take last names, and in a lot of cases, laws saying that Jews, specifically, be given last names. So, government officials went around assigning last names. So, a lot of central European Jews got their names that way. Berg means mountain, Burg means castle, and Stein means stone. A lot of times these names were just given out randomly, according to the namer’s whim.
Cohen is a different case, as is Levi or Levy. One of the 12 tribes of Israel was the tribe of Levi, and a subtribe of the tribe of Levi was the subtribe of Kohen. Back when the temple still existed, the tribe of Levi made up the temple workers, and the subtribe of Kohen made up the temple priesthood. Even though the temple isn’t around anymore, Levis and Kohens still have a special place in Judaism. So, somebody with the last name Levi, Levy, Levine, etc., is probably, although not neccesarily, a Levi, and somebody with the last name Kohen, Cohen, Kahane, Katz, etc. is probably, although not necesarily, a Kohen.
The word “Jewelry” doesn’t have anything to do with Jews. It comes from the French, and probably ultimately from the Latin “iocus”, which is also where we get the word “joke” from.