It teleports a small bit away, so most likely it landed somewhere on the island...but depending on how the island generated that could be some underside ledge or random hole in the ground that sometimes generates (or was carved out by you and not filled back in when you arrived).
It is possible for the egg to teleport onto a portal block, and since you can't stand on a portal block the egg would then go to the overworld spawn point (because items and endermen cannot sleep in a bed to change it).
EDIT: oh, yeah, if your entry platform is buried into the main island and is close enough for the egg to land on the obsidian...if you then left and came back when the dragon egg was still on the platform then it's gone forever. Everything 3 or 4ish blocks above the platform is automatically deleted to ensure the player never ends up suffocating in placed blocks when entering the end.
no, the dragon egg never overwrites blocks (outside of carefully-designed contraptions/farms that make use of duping glitches, but those involve generating lots of redstone ticks, lag, or being far enough away to invoke lazy-chunk mechanics that can interfere with normal entity processing).