Are my texts sent to DeepL every time I use the Ctrl+C / ⌘C keyboard shortcut?
No.
When you copy a text (with Ctrl+C or ⌘C), nothing will be sent to DeepL. Only when you press Ctrl+C / ⌘C the second time will your text be sent to DeepL for translation. If you are logged in to the app with your DeepL Pro account, your texts will not be saved.