Imagine that every imaginable moment in every reality has either already happened, or is fictional and never can happen. A Szierkhogh exists across all moments which, for it, are real. Which ones those are depends on two things: The Szierkhogh's constraints, and its nature. The constraints determine what it is permitted to do by its superiors, and its nature determines what it is inclined to do. A Szierkhogh is incapable of acting outside of either set of parameters, but exists within all moments in which it behaves in accordance with both. Because it is not constrained by time or space, it is aware of every state in which it exists, from every state in which it exists (barring limitations imposed by its constraints). A Szierkhogh can exist in degenerate timelines, such as single isolated points, or ones which would have arisen from fictional timelines. They are usually permitted to imagine fictional events, but as these tend to be distateful to the Szierkhogh, doing so in most cases would be outside of their nature. 
They are perfectly capable of observing whether something is endless. As an example, if it is in their nature to participate in something embedded in time without ever stopping, then they can confirm whether their existence has no end along a path in which they attempt to do so.
Each Szierkhogh has a mechanism assigned to it which, when activated, does the closest thing to truly killing it, which is essentially making it fictional to its superiors, and them fictional to it, from that point forward. Obviously, Szierkhogh are not permitted to interfere with this mechanism, but they are typically also not allowed (nor inclined) to do things which would cause it to be used.