Do the House Elves share the family name (aka last name) of their masters? If so, what happens if they are freed? For example is Dobby "Dobby Malfoy" or is Kreacher "Kreacher Black"?
If they do not have last names does that mean they don't have actual families? If they do not have families where did they come from and how were they born?
No house-elf is ever referred to by more than a single name in any canonical materials - books, Pottermore, or even films. Dobby, Winky, and Kreacher are always called simply Dobby, Winky, and Kreacher, not Dobby Malfoy, Winky Crouch, Kreacher Black, or by surnames of their own.
According to Wikipedia (which is of course not a canon source, but it's the best I could find):
House-elves are small elves used by wizards as slaves. They are 2–3 feet tall, with spindly limbs and oversized heads and eyes. They have pointed, bat-like ears and high, squeaky voices. Their names are usually pet-like diminutives, and they do not appear to have surnames. They habitually refer to themselves in the third person.
(emphasis mine)
But we don't really know much about house-elf culture, if indeed they have any culture among themselves beyond simply serving wizardkind. So it is possible that house-elves have surnames which they might use to each other, but the wizards they serve simply don't care and refer to them by a single name because it's easier. (I've previously discussed the issue of house-elf culture here.)
They certainly don't have surnames of their own:
He wanted to ask, 'What are you?' but thought it would sound too rude, so instead he said, 'Who are you?'
'Dobby, sir. Just Dobby. Dobby the house-elf,' said the creature.
Harry Potter and the Chamber of Secrets - p.15 - Bloomsbury - Chapter 2, Dobby's Warning
Of course, it is still possible that they take the name of the family they serve. In this case, presumably Dobby wouldn't want to reveal his surname to Harry, which would still be compatible with the way he introduced himself to Harry. All the same, in that case, it seems to me that you wouldn't introduce yourself as 'just Dobby', you'd just be careful not to mention the name Malfoy. But he clearly doesn't have his own last name.
House-elves so do not share their family's last name. As @anemone posted in their comment, that would be downright shameful on the family. House-elves are considered a family's belongings, as any interaction between the Malfoy's and Dobby can prove this.
However, there is an explanation for this. There is no need at all for house-elves to be documented, and therefore they do not have last names.
Some might argue the House Elf Legion Office dealt with fired house-elves and whatnot, but this clearly is not the case. When Dobby and Winky were freed, there was no department or office that they went to, at least not explicitly mentioned.
"When Dobby goes to Winky, and he finds out she has been freed to, sir! So we goes door to door looking for work, and many slammed doors in our faces, saying 'House Elf-looking for pay," they says, "that isnt the point of a House Elf."
They would have visited said department, if there really was one. It sounds as if the Ministry is trying to seem on top of any possible issues to satisfy the public.
Does that mean they don't actually have families?
Dobby seems to be a relatively young house elf (they have life expectancies of several human generations) and each Elf's parent seems to have passed by the time their offspring comes of age. (See Dobby and Winky)
Also, how JKR has lead us to believe the reproduction process of house-elves is a house elf from one esteemed family and one house elf from another prominent family, (ex. Malfoy and Black) and the offspring lives with the mother, who raises it. When the offspring comes of age, the mother appears to either die of old age or be "taken out back" by the owners.
See quote:
This is the heads of beheaded house-elves, when they got to old to carry a tea trey.
That already isn't a proper family, so its safe to say that few (free elves) to none had proper families, if you consider a proper family to be a Nuclear family. (Nuclear families were the 'norm' in this time period.)
Elves did not have proper families, and also had little documentations, if at all. Therefore, there wasn't a reason to give a house elf a last name.
I don't think they have last names because they aren't really humans. And they aren't born into those families are they?
