Why is the rate widget (from the Rate module) not showing up?
From the screenprint about "edit widget" in your question it shows that you've created a "Yes / no" widget using the Rate module.

But nowhere in your question it shows for which "Node types" you've enabled your widget. Since you're using the "Yes / no" widget (located at admin/structure/rate/add?template=yesno when adding a new widget), you should indicate the desired "Node types" there for which the rating widget is to be enabled. If you didn't specify (select) any of node type, then that would explain why the rate widget is not showing up".