Nicolas Cage ate a real cockroach for this film - it reportedly took three takes. He once said about the experience, "Every muscle in my body didn't want to do it, but I did it anyway."
Peter's strange accent is supposed to be a fake accent used by the character because he thinks he sounds more "elegant" and "smarter" (hence the literary agent job). The accent comes and goes throughout the movie and is more prominent when trying to impress people, and less in scenes with people like the psychiatrist.