Just in case the OP hasn’t gotten the message, family names have not always been a part of human culture, so Adam and Eve (had they existed) most probably would not have family names.
In fact, the use of family names, as a rule, in European culture is relatively recent. It did not become de rigeur in Britain until about the 15th century – so only about 600 years ago. In parts of India, family names are either not used at all or have been instituted only within the last 50 years. In many places in the world (such as Indonesia and Iceland), family names still are not mandatory.
As has been stated before, there are four main sources of family names – patronymics (parentage), occupational, descriptives, and place names. Things like clan or tribe names usually can be pegged to one of the others, often patronymics.
Regarding names like “King,” it is important to note that it is not likely that the first person to hold this name was an actual holder of this title. Most likely it was someone who held an (often menial) job associated with the king. Or it might be some kind of descriptive or nickname for someone with a kingly manner. “Pope” is likely a nickname for a Catholic, rather than an actual pope (who might be unlikely to have acknowledged any offspring who would inherit such a name).
As to terminology, this is a very important issue when one is speaking of names. Generally, the American terms “first name” and “last name” and the British terms “Christian name” and “surname” are all considered imprecise and are disfavoured by genealogists.
After all, it doesn’t make sense to speak of “first names” and “last names” when Chinese, Japanese, Koreans, and Hungarians, for example, put their family names first. And it doesn’t help when you’re talking about a culture, such as Iceland, where individuals do not inherit family names (rather, they have patronymics). Also, in places like Indonesia, people often have just one name. In many Arabic cultures, a person might have multiple names, but none are inherited like a family name is inherited.
Similarly, the term “Christian name” only makes sense when you’re talking about someone who has received a name as part of a Christian ceremony, such as baptism.
“Surname” is also a specialised term for a name that has been “added,” such as William the Conqueror.
The preferred terms are, thus, “given name” and “family name.”