The word “Elf” seems to have originated in Scandinavia, specifically among the Norse people of Denmark, Norway and Sweden and spread to Iceland and Greenland with their settlers.
Norse mythology in general is quite fascinating. Some great original source material for learning about Norse Mythology is the Prose Edda (the Younger Edda), the Poetic Edda (the Elder Edda) and even Beowulf.
JRR Tolkien drew heavily from Norse Mythology in creating Middle Earth.
In Norse Mythology, Elf and Dwarf are sometimes interchanged. There are the light Elves, Liosalfar and the dark Elves, Dockalfar, (in some places they are referred to as the Svartalfar as well as the Duerger and the dark Dwarfs.)
Since sources conflict, there is some confusion between Elf and Dwarf, but in the nine worlds of the World Tree, Yggdrasil, the elves and the gods live in separate worlds, Elfheim and Asgard, respectively. The Aesir, war gods, fought the Vanir, nature and fertility gods from Vanaheim, long ago and won. The Vanir sent hostages to live with the Aesir, two of which rule over the Elves: Frey and Freya.
Similar in morphology to these Norse Elves, the Fairies of Celtic Mythology are the old gods, driven underground usually into burial mounds by the encroaching invaders. In Celtic countries: Ireland, Scotland, Wales, Britain, Brittany, and France (known as Gaul when it was primarily Celtic), the Fairies, once underground, shrank in stature. In Arthurian legends, potentially from the 300's AD, but mostly attributed to the 1400's and France, the Fairies are still full size humans and objects of temptation. But by the 1800's the Fairies have become the Wee Folk of the Leprechaun and Flower Fairy variety. William Shakespeare uses both sizes in his A Midsummer Night's Dream, written in the 1600's.
Does the Leprechaun's shoe making remind anybody else of the Dwarfs' mining and smithing? Maybe it's the hammers.
This shrinking in size of Fairies and Elves is most likely thanks to the effects of the Enlightenment. (Beyond a doubt, Science greatly improves standards of living, but sometimes it can rob the imagination of its nobility and power for both evil and good.)
The Nymphs of Greece and Rome seem to many to share a similar morphology to the Elves of the Norse and the original Fairies of the Celts. While the Fauns and Satyrs, though not sharing the size nor the use of hammers, seem to to me to share similarities with the Leprechaun and Dwarfs in function: being comical for their drunkenness and lechery, and form: being hunched and shod with a noisy shoe. (Fairies and Elves being mostly shod in slippers or barefoot.)
Perhaps the Vampires of the Gothic Mythology shares some similarities as well with the Fairies, who are sometimes composed of dead relatives. Then there are the Rushkalas of Russian Mythology, also related to the dead and of a pale complexion. And the Fox Fairies of the Japanese Mythology seem not unlike the Nymphs of the Greeks to me, though the tails are more like Satyrs and Fauns.
So Elves, when representing the old gods, nature spirits, and the dead that live in the wilds or underground can be tall when used as fertility symbols and when used as good luck, comic or evil symbols can be short or even tiny.
So how do Santa's Elves fit into all this? Well, because Christmas Elves were meant to be comic for children and definitely not fertility symbols they are short. And since they make toys for children they take on more dwarf like characteristics with hammers.
This distinction between tall noble Elves and short, comic, hammering Elves even gives Tolkein difficulty in reconciling. I highly recommend reading, for yourselves or to children, or listening to on an audio book, Father Christmas Letters, by J.R.R. Tolkein. It is a collection of letters Tolkein wrote to his children from Father Christmas. Here Tolkein has Christmas Elves and they are different from his Middle Earth Elves. It is also a great read at Christmas time for the whole family.