When I look into European dragons, most of the culture around them is either in the form of heraldry or as the monsters killed by heroes (mainly St George, Siegfried/Sigurd, and Beowulf). Is there really anything else to look into for the European conception of a "dragon"?

This is a really interesting question, and I had to chew on it for awhile. You did well to describe at the outset how dragons appear in heraldry as well as early medieval literature - two separate manifestations with little to do with one another. The beast is perfectly suited as an opponent against which the hero can demonstrate courage and physical prowess. From those medieval sources, we can glean something of what many Europeans once thought of dragons, but somehow, that's decidedly unsatisfying.

Dragons, like giants, are entities that the pre-modern European folk believed once existed - or existed far away. Because of this, they occur in historical legends - stories told to be believed about past times. One does not normally find dragons in legends told about the "present day" - stories told to be believed about what our contemporaries (or we) have encountered: as in "I once saw a ghost..." or "I once saw a UFO..." One would not hear a European say, "I once saw a dragon..." or "I once saw a giant..." They would say, "Long ago, there were dragons..." or "Dragons live far away..." The Anglo-Saxon Chronicle describes flying dragons in the sky as an omen of bad things, but it is not clear if the author had seen them and made that interpretation or if he had heard of them - and it is possible that the "far off" idea is still present - but that aside (we needn't return to early medieval sources!!!).

Because of the tendency not to hear about dragons in contemporary legends, in the pre-modern folklore they tended to be restricted to the folktale, stories told as fiction, the novels of the folk. The dragon manifests, famously, in the folktale known as ATU 300, the Dragon Slayer, which manifests in the Greek myth of Perseus and Andromeda. In many nineteenth-century collected examples of this folktale, a giant or ogre has replaced the dragon. Indeed, many compare the original Star Wars film as an expression of the rescue of the princess from the dragon, and here the dragon manifests as a guy in black clothing with a black cape and black face-covering helmet. So much for the dragon!

In considering an answer to your question, I consulted the enormous Motif Index of Folk Literature by Stith Thompson. He lists all sorts of motifs associated with various things, gleaned from folktales, largely of European (but not entirely) origin. Nothing leapt out at me, but I was struck by one thing: his index lists various motifs for dragons, but the entry is less than a page long. The one for giants in folktales is two pages long, and there is another page for ogres. This is hardly scientific, but I believe it does point to something: the pre-modern folktale as collected by so many throughout Europe beginning in the early nineteenth century tended to employ gigantic human-like creatures in place of the dragon employed by medieval literature. And the thing about giants is that they are usually, really dumb, so the hero can outsmart a giant, but outsmarting a dragon is usually not an option. Both are big and deadly, but usually, there is more physical prowess involved in killing a dragon and pre-modern folktales celebrated brains more than brawn in its heroes. That's a gross generalization that would be easily disproved by specific examples, but I believe there is a general trend, and perhaps that made the dragon less appropriate for many pre-modern folktales where a giant was a better opponent.

That doesn't help much to answer your question, except that my initial response was to tell you to look to folktales for examples of dragons, but when I looked, I found that the material is thin.

We can make one observation about dragons: they tended to shift from a water-based, venom-spewing serpent that was more sea-monster to the modern mind; and this gradually was supplanted, transforming into a winged, flying, fire-breathing dragon by modern definition. Perseus's "dragon" was in the sea, and Siefried's "dragon" was a slithering "worm." Flight was uncommon for the early references to the European dragon, and yet it is essential for more recent references and for the modern image of the dragon. As dragons took flight, they found a home in heraldry more than the folktale!

I believe with many words, I have been of little help to you!