I have an class with event listeners wired for dragging the object. I want to
add an event to this class to delete the object from the stage.

I create the object like so:
private function makeItem():void {
var foo:ItemImage = new ItemImage("assets/model810.png");
picArea.addChild(foo);
}

I know I can remove it from the parent app with picArea.removeChild(foo), but
how do I do it from with the class?

If I add an event listener like:
this.addEventListener(MouseEvent.DOUBLE_CLICK, deleteMe); inside my class,
what's the command for the instance to remove itself?

Makes sense, yes. Is there no way to trigger this action from the child?

My issue is that the main app is going to spawn my class, add the image path
to it.

The class has code to allow it to be dragged around, etc. If there user
decides he wants to "delete" the object he added, I was hoping to perhaps allow
for an "option-click" or something similar on the object to remove it.

How would you recommend I do that?

That button should broadcast an event so the parent can react by removing the child.