Should I keep copies of my downloaded files?

Yes — treat your own computer as the system of record:

• Keep your final exports. Once you've unlocked and downloaded the clean cover PDF, interior PDF/EPUB, mockups, and so on, save them somewhere permanent (cloud drive or a backed-up folder). You can always re-export from BookPublish, but having the finished files in hand is smart.
• Keep a project-file backup. Download the project file so you can rebuild or revise the book later, even on a different computer.

Your browser storage is a convenient workspace, but it can be cleared by you, your browser, or your operating system. The copies you save are what make your work durable.