Prince Prospero is a "happy, dauntless, and sagacious" person. He is a flat character because he doesn't evolve from being close-minded. He also is a static character because he doesn't change from being careful or fearful.
Death is a flat character. He doesn't change at all, and he's also a static character. All Death does is kill the people that Prince Prospero tried to protect in the castle.
After Prince Prospero looks in the mirror, he went to find who had mocked the Red Death. Once he found the culprit, he had him unmasked, only for it to be revealed as Death himself.