Nix works by putting its packages into some installed folder, and then building a system by symlinking to those packages.

You'd retain the nix package storage between boots (so as to avoid having to redownload/rebuild), and setting the system up just involves setting up some symlinks. - GrahamC's blogpost discusses this in a bit more detail.

"Fresh system each run" isn't too exotic compared to image-based approaches like launching a VM on AWS, or running a Docker image.

No, since all your system is declarative all the software and related configuration is redone at each reboot. Just the state is removed.

To avoid losing data you want to keep you can also mark directories or files to be stored in the permanent storage.

Take a look at the nixos-impermanence that automates this task to you.