Furthermore, many myths about witches (flying around, the ability to transform into animals, the ability to use magic) seem very similar to near-universal traits associated with indigenous shamans all over the world. Were these myths simply carried over from Europe's shamanic past, or were they "re-discovered" after the fall of the Roman Empire?
I realize these are broad questions, but I suppose I'm just trying to understand how much of the "witch" stereotype is based on reality, and how much is totally fabricated.
Europe had traditions of both good and bad magic before, during, and even after the witch trials of the late 15th to 17th centuries. The "innovation" of the late Middle Ages is the diabolicization of maleficium or witchcraft: it becomes the act of consorting with the devil in order to work harm. A surprisingly uniform stereotype of "the witch" is put together in the 15th and 16th centuries out of various elements of prior and current folklore, theology, and practice (scholars working on masculinity & witchcraft have found roots of the witches' Sabbath idea in contemporary village festivals, for example).
I can't say much about comparative witchcraft studies beyond Europe. But I've addressed your question about a relatively new stereotype here, if you're interested: