An island is mostly rock, so if it didn't go all the way down it would sink! The exception is ice-bergs, which do float, ice being less dense than water. Look at a map of the Earth, and you'll notice that there's no land marked at the north pole. There isn't any - it's just a floating mass of ice.

The area beneath these floating mats is exceptionally rich in aquatic lifeforms. Eventually, storm events tear whole sections free from the shore, and the islands thus formed migrate around a lake with changing winds, eventually either reattaching to a new area of the shore or breaking up in heavy weather.

Floating islands are a common natural phenomenon that are found in many parts of the world. They exist less commonly as an artificial phenomenon. Floating islands are generally found on marshlands, lakes, and similar wetland locations, and can be many hectares in size.

Hypothetically, yes, but it would need to be perfectly balanced otherwise the island would simply flip over and the repelling force would become an attracting one. Causing it to crash to the surface.