An Elf with shapeshifting ability? Why?

I have read the tale of Beren and Luthien, and it might have slipped past me; but how is it that Luthien, out of all the Elves can shapeshift? Is Melian capable of that ability? If so, it has escaped me also. If I remember correctly, Tolkien does not really give any background to the nature of her power. It seems that Luthien... just has it.

Also, I thought the ability to shape-shift was a power reserved to the fallen Maia such as Sauron. It just seems that Luthien was using an evil power to counter evil.

Even so, since Luthien had this ability, and through it gained access even to Morgoth himself, why did she not attempt to assassinate him then and there? They could have slain him AND taken the Silmaril. It just seems weird how Tolkien elevated Luthien into an elf who seemed to possess a power with more potential to defeat the evil which was Morgoth than any other high Elf, including Galadriel. I have not seen too many people speak of Luthien's power much, and I guess my main question would be on the specific and peculiar nature of her power.

Best regards