How can I delete or modify the original image uploaded?

When deleting an original image, all of its derived images (transformed versions) are deleted as well. How to destroy the original image and all it's derived versions can be found here.

To modify the original image, Cloudinary supports incoming transformations, this will transform your image before storing it in your account. Here's a PHP code example:

\Cloudinary\Uploader::upload('/home/my_image.jpg', array("width" => 1000, "height" => 1000, "crop" => "limit"));

Then you can either deliver this transformed image as is, or create any further transformations that you require (e.g., thumbnail). 
The original image will never be stored in the first place.

Upload API's documentation can be found here.