Is it possible to automatically backup cropped images?

Am I able to issue an API request which will create a new file (with a new uuid) that is a cropped image of another file, so that the automatic S3 backup feature would capture both the original and the cropped image? In other words I would like to have 2 files I can get at urls /uuid1 and /uuid2, where uuid1.jpg is the original image and uuid2.jpg is the cropped image.

This is absolutely possible. You can create a copy of a file with crop operations applied via POST request: https://uploadcare.com/documentation/rest/#file-post

Source parameter will be your file cdnUrl. If you want this file to appear on backup bucket, you need to store it. You can do it via separate store request for new UUID, or you can add store=true parameter to copy request to do it automatically (this is not mentioned in documentation yet).