Ender Dragon characteristics

If you go far enough from the island, the health bar for the dragon will disappear. It is safe to build at that place. See for example this video at 08:30.

Once all players are outside a 320x320 square centered on the dragon, the server will stop updating the dragon entity (time stops for the dragon). Otherwise, the dragon can go practically anywhere in The End.

The Ender Dragon has some unusual characteristics. The Ender Dragon can spot a player practically anywhere in the Ender. However there are some set backs. Your mob spawner must be out of obsidian because the Ender Dragon can destroy any other material it comes in contact with. It will look like the TNT explosion. The Ender Dragon usually will hang around it's crystals for "Power and Food" but will swoop down on a player if the are within 30 blocks of a crystal. The Ender Dragon's flying radius is huge but it's hard for him to turn easily. He has to make large swooping motions, about a 40 block diameter for it to come around and attack again.

The Ender Dragon characteristics I've seen are that he sees me no matter where I am at on his island and if I stay about 20 blocks from his towers he will only attack me by coming straight at me and trying to ram. Only when I'm near the towers does he toss magic at me. If I stay 20 blocks or so away and shoot at him as soon as I see him plowing through Endermen I can score 4 or 5 shots per ram attempt and he starts to turn around at the third shot I score.