Go to sinodefence.com and look it up, the missile section. You will find that there are indigenous chinese supersonic missiles.
And no, speed doesn't necesarrily mean a missile is better than a slow one. But if same guidance, tracking and targeting tech is used, then yeah, the faster one would probably be better. Disadvantages would be bigger heat signature and somewhat smaller potential manouverability. Real edge of speed becomes obvious when a salvo is fired, and ship defences just dont have time to coordinate their their attacks on every missile coming towards them.