Mixing Polymer 1.0 and Angular 1.x
If your aim is to use Material Design (which for web based apps boils down to using the correct themes), I just saw Angular Material (https://material.angularjs.org/latest/#/). Maybe that would help?
The first problem can be solved by using
window.Polymer = {dom: 'shadow'};as stated by Justin Fagnani in the comments, with slightly different syntax than stated in the polymer docs, otherwise it will not be recognized in Firefox/IE, see https://github.com/Polymer/polymer/issues/1844
However, I guess that means that you won't get any performance improvements that were introduced with shady DOM.