To hatch the dragon egg, you need to earn the invisible shield. Once you do this, it will allow you to go to the egg, and hatch it. Otherwise you will likely be killed with every attempt.
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That is just a block, you can`t hatch it YET.
It is not yet possible to hatch the dragon egg in Minecraft, the egg is like a trophy to recognise that you have slain the Ender Dragon. Happy Minecrafting!
No they don't hatch.
Currently in the Vanilla release, dragon eggs do nothing, and are primarily used as an ornament.
The ender dragon drops it when defeated