Sensitive data like passwords and cookies are never read or stored. Notyfy sees which of the available platforms you're logged in gets the notifications from them. You will never be asked by Notyfy to login additionally.
Notyfy only requests the needed platforms for notifications without loading images, ads and trackers.
Notyfy will periodically check for notifications of platforms that you are logged in to. Since this request is coming from the Chrome Extension through your browser, the requests will be identified as you and is able to check your notifications on your platforms.
The requests are super lightweight (from 0.7kb to 58kb) compared to, for instance, the ~25 mb Facebook page load.
Nothing is stored on servers. The only thing being locally stored for caching purposes are your notifications for a faster loading time.
No, the cookie permission is not required.
Yes, it's available here: Firefox Add-on