Not if you are already in the European Union. If you enter the U.K. From outside Europe through the Isle of Wight then, yes, you would need a passport to get in just as you would to get in at Heathrow. Obviously, if you came by boat or private plane from China or Africa and landed on the Isle of Wight secretly at dead of night, you might get away with not having a passport, but you would be in the U.K illegally.
No you don't at all, just a boarding ticket (obviously)
To come into the United Kingdom you need a passport. However, once in you are free to travel to the Isle of Wight without showing your passport.