One thing to keep in mind when it comes to heuristics, is that they should be considered as "broad rules of thumb" rather than strict guidelines that should be followed at all times (Jakob Nielson says so himself).
Instead of blindly following those heuristics, we should therefore ask ourselves: Is this information of interest or relevance to the user?
One simple example where this applies is the system information of our operating systems: The OS could display memory load, clock cycles, core temperature, load balancing management, disk fragmentation etc. at all times. This could be seen as an exemplary fulfillment of the first heuristic, but in reality, it would most likely just overwhelm the users and waste a lot of screen real estate. It is therefore reasonable to hide this information from the basic interface. The portion of users who are interested (and sufficiently skilled) to find the data, can still access it, but they actively have to do so - the information is not imposed upon them.
The same philosophy applies to your concrete questions: I would argue that Facebook or StackExchange notifications are integral part of the main functionalities, respectively, and most users will want to see them. Therefore it is reasonable to keep them always visible - otherwise it would be a violation of the first heuristic.
For your question regarding non-real time operations ("request awaiting response"): Ask yourself (or find out) if/how this information is relevant or important to the users. For example, if we are talking about a parcel tracker, the information about what is going on with my delivery is arguably the most relevant, even if the status might not change for days or even weeks. On the other hand, if I'm reporting a cosmetic bug to an otherwise functionable website, I might not be that interested in the status of my request. In that case the information does not have to be that visible, and could easily be "hidden" in the corresponding subsection of the website.
One final note: The heuristics can rarely be seen in isolation, and one heuristic might influence several others. We could strive for perfect fulfillment of the visibility heuristic, but that would most likely harm the aesthetic and minimalism heuristic, for example. We should strive for a good balance of those different considerations. Our main rationale, after all, is not perfect rule abidance but to provide a good user experience.