I understand I can get the scope by element:
scope = angular.element($0).scope();
scope.$id; // \"003\"
How do I get reverse: Find the DOM
You can always get the scope using element's id.
Example:
html:
js:
var myEl = angular.element(document.querySelector('#myId'));
var myScope = angular.element(myEl).scope();
Live example : http://jsfiddle.net/choroshin/7XQA7/2/
also as David Chase suggested, you can always use batarang - Extends Chrome Developer Tools, adding tools for debugging and profiling AngularJS applications.