How can I upgrade the stack immediately after a new update? Do I need to consistently check the changelog, reinstall the whole stack and migrate the store's files and database?
There's no way to automatically upgrade the stack. When you want to upgrade the whole stack, you will need to launch a new image and migrate your site there.
Also, does the instalation under "/opt/bitnami/stack/prestashop" contain anything different from a regular instalation of prestashop? Is it safe to just replace those files with another local hosted prestashop instalation of the same ps version?
You can try, but we don't guarantee that is going to work.