For a plane that can truly (albeit briefly) fly backwards, checkout the Sukhoi-30. It is a Russian fighter jet with truly vectored exhaust - think F-22, but 2 axes vs. 1 axis. Here is a video of it doing loops and Cobra maneuvers:
What are the advantages of such maneuverability? I can't believe they're trying to dodge missiles... Is it for aerial combat purposes?
Modern missiles cannot be dodged. It's mostly just a way to show off the plane's agility. It could just barely be useful for shooting at a target behind you, but you sacrifice a lot of energy moving your aircraft like that.