We have a small amount of back story of the Giant Spiders, who were descendants of Umaiar (Fallen Maiar), but not for the want of looking I can't find anything relating to the origin of Dragons. We know that Ancalagon was a servant of Morgoth and the dragon race were inherently evil (for the most part). Are they twisted from another race? Or did Eru breathe life into these creatures and their nature is just generally filled with greed, pain and suffering?
Like many other things in the Legendarium, there's no concrete, canon answer for this. Tolkien didn't specify it in the published books, and his notes tend to contradict themselves as his conception changed over the years.
The entry on the origins of dragons in the Lord of the Rings Wiki does a rather thorough job of bringing the various theories, their strengths and weaknesses:
- Dragons are beasts that were physically enhanced by Melkor, perhaps corrupted giant Eagles. This can explain how he "made" them without bypassing the limits of Creation, but doesn't explain how he gave them intelligence, hypnotic powers and fire-breath, nor does it acknowledge that the original dragons were, in fact, wingless.
- Dragons are sparks of the spirit of Melkor himself, given physical shape. I don't know exactly what the motivations for this theory is, because the Silmarillion has nothing even remotely implying this. It is said that Melkor bred them, not spawned them.
- Dragons are magical creatures created by the Eru and the Valar in the Music of the Spheres, just like they created all other plants and animals, Ents and Giant Eagles included. They may have had a different original purpose by Yavanna, but they were drawn to Melkor, corrupted by him, and bred to be larger, flying and more vicious.
- Dragons are Maiar, in an earthly form, much like the Balrogs. This makes sense, gives them an understandable source of their powers and might, but doesn't explain how they breed - dragons are expressly stated to have children - "Glaurung and his brood" - which doesn't fit with the Maiar concept, since they're not an actual race.
- The final theory, which you bring yourself in the question, is that they are the children of Maiar and natural beasts. Much like Ungoliant and the spiders, so have Maiar, at the behest of Melkor, bred with natural lizards and from them spawned the race of dragons. It makes the most sense to me - we know Maiar can breed with other races, either spiders or Elves (Melian and Thingol), and we know that their offspring tend to be more powerful, beautiful or meaningful than others.
Dragons were a race of monsters bred by Morgoth. It is my understanding (from all the various material I've read, including the HoME) is that dragons were originally some beasts or serpents that were corrupted and transformed by Morgoth. http://tolkiengateway.net/wiki/Dragons