And why is this? Do they all change their names, meaning that Prince Charles will be King something-else?

Yup. Just like the Popes the British royalty have the right to call themselves what they so wish. The current queen was in Treetops when she heard the news of her father's death and returned by plane. She was met at the airport by a royal dignitary who asked her what name she wished to be known by. Why? Elizabeth, of course!, she replied.

Is there some type of rule about what a royal family member can be named? William is pretty normal, too. What if William named his son Ethan, or Austin. Would they allow a King Austin?

There are no rules per se by which a royal may name their child. Tradition tends to dictate that and the royals are no different to any other family in the world who tend to hold on to family names. Elizabeth was named after her mother and was the first prominent royal to bear that name since Elizabeth R. back in the mid-sixteenth century.

George and William were the common names for the Prince of Wales for the past two hundred and so years and only Prince Albert (who became George V) and the current inmcumbent of that title (Charles) have been the exception to that tradition.

William is now making a comeback through Charles' son and Prince Henry (Harry) will be the first Harry since Henry VIII to hold the title Duke of York (yes, Henry VIII was Duke of York as his elder brother held the Prince of Wales title until his early death).

Had Charles not married Diana and had two sons Andrew would assume the title after him and then the lineage would pass down via his two daughters, Beatrice and Eugenie. There has never been an Andrew on the English throne so he would probably be advised to take another titular name.

As for Ethan or Austin I don't think that a Prince would be given such a name. There is a strong sense of tradition in a tightly knit family like the Windsors and they probably have a bandobast before deciding on a final handle which probably comes from a list of names decided upon and accepted by the monarch. After all, it is very unlikely that the current monarch would like to be followed by a King Wesley or a Queen Kylie a few generations down the line. But Harry Pothead may be the first Royal rebel to turn all of this on its head and throw such a spanner in the works.