Adding different classes to different images depending on size for fine control of image layout in posts

It looks to me that those images, along with their divs, have been added directly into the editor. So, it’s probably being updated by a developer, or at least someone HTML savvy.

I’m not sure what you mean by ‘meta boxes’. A meta box is basically any collapsible box within the WP admin interface. If you actually mean custom fields, then there are many plugins for adding images to posts. Take a look at Advanced Custom Fields, as just one example.