Is there a way to create CSS image styles? Is there a way to assign css to image styles?

I don't see any drop down for custom...

Would be nice to add img-responsive style to ALL images on the site...

Is there a module that can do that? Shouldn't it be native functionality?
What you are looking for is considered theming and is done mainly in the theme's template.php file. 

There is a guide available here: Add bootstrap image css class to pictures 

&lt;?php
/**
 *
 */
function THEMENAME_preprocess_image_style(&amp;$vars) {
        $vars['attributes']['class'][] = 'img-responsive'; // http://getbootstrap.com/css/#overview-responsive-images
}
?&gt;