I'm old enough to watch this show in first run, but I was young enough to be excused for liking what I now think is a pretty awful cartoon. Incidentally, I also developed a dislike for roller coasters...
The DM was a character in the show and appeared at the beginning of the episode to give the kids some cryptic message, then often appeared at the end of the show to deliver the moral to the audience, though the kids never saw him at the end of the show.
I was bothered by three of the characters, even as a kid (though, I though the other three were pretty cool). In order of least annoying to most annoying, the little barbarian, the cowardly cavalier and the bumbling mage. Grrr. They were such cliches, even then, it really botered me that they were applying those cliches to D&D.
I had forgotten all about the unicorn. It annoyed me, too. In general, I've always been annoyed by the tacked-on characters that are with the group only for a cutesy or comedic purpose. Snarf from Thundercats, and Orko from He-Man and the Masters of the Universe come to mind as other examples. Oh yeah, and the worst of them all - Scrappy Doo.
I've attached a pic of the Dungeon Master. (Click to see full-sized.) He was about two or three feet tall.