I'm kind of confused about what circumstances causes the Demon to show up in any given song-level and throw a boss at you. At first I thought that he only appeared in the Symphony Soundtrack, but I've played through every song and I haven't won the game, so I guess there's more to it than that. Is there some way I can improve the odds of entering a level with a boss in it?

From what I've seen, their appearances are pretty random. They won't appear for a couple of tracks if you've just fought another demon, though, and they won't appear while you have a "captured" song.
Also, they certainly appear on non-built in tracks.

My impression is also that longer songs make him more likely to appear, as the boss battles require a certain time for the countdown. But maybe I'm wrong and that countdown is already considered in the minimum requirement for song length ...

Well, I've seen a demon appear in a song that was barely longer than the boss battle, so I guess the only requirement is that the song is long enough for the battle.

Ah, in that case I guess the boss battles are the reason for the minimum song length requirement and a boss could theoretically appear in any eligible song.

Okay. Do you need to change up your artists, or can you get more than one demon from the same album? Or even from the same song?

Also, does the difficulty matter? Can you still encounter the Demon if you're not playing on the highest available difficulty?