I think that the sparks are "simply" geniuses in the old sense. The idea is that great artists cannot teach you how to create works like theirs, because their secret is simply that they have the genius, and they cannot pass it over to you. It is an innate power, that allows them to see something invisible to the others (either how Nature works, or the Absolute, depending on whom you ask). However, their works can serve as model for a school, and thus represent a breakthrough for artistic production in general.

Sparks have this sort of clarity with scientific matters: once they have an objective, they can reach it because they understand how physics etc work in an immediate way that they cannot teach to others.

The fugue state is very much like the "madness" of the artist as he performs. This is also a very old theme. I don't think it's random that the Baron and Gil look a lot like Beethoven when they first appear, or that Agatha plays music when she destroys the Baron's men.