Does Dropbox Carousel have to be Open to Auto Backup on Mobile Device?

No. You don't need to be open a Dropbox Carousel to Auto Backup photos and videos on mobile device. It has "Background Uploading" function which helps your Camera Uploads finish without the need for you to open the Carousel app. When you are uploading a lot of photos and long videos, you may notice that Carousel stops uploading shortly after you close it and resumes uploading only when you open it again. This occurs due to a time limit iOS puts on apps running in the background. With background uploading function, Dropbox Carousel will automatically resume uploading for a few minutes whenever it detects a significant change in your iPhone's location. The Background uploading only applies to iOS devices.

If you have chosen cellular data, then Carousel will periodically upload photos and videos without needing you to open it. If you have chosen Wi-Fi Only, then it will resume when your device connects to a Wi-Fi network again. Background uploading will not use your cellular data.