A sandbox is an application-level security feature meant to prevent exploits and data leakage. It’s about preventing your browser and OS from getting hacked.

The cookie thing is a restriction about cross-site tracking. Normally, cookies are restricted by being read by the domain that set them. However, this was abused by marketing tech, where they would embed code from Facebook or whatever on different sites, and then read them on any site that had that code. The Total Cookie Protection makes it so when a third party sets a cookie on a site, it can still only be read from the main domain of the page it was set on, preventing cross-site tracking.

No, it's just like ... deleting cookies after every interaction?

If a login was simply cookie based (which it is usually not) you would not be able to login at all, if another mechanism is used like e.g. storing user / password in in a backend database, it will have no effect at all.

Cookie protection is more aimed at cookie- based tracking, which works on most websites even without any login.