Is "no arising" a thing or just something we may be taught? If it's a teaching only, I guess it's used to show that something else is not the case (showing e.g. that series do end).
If it's a dharma in the other sense, then does it exist relative to things arising, so that something arises whenever another thing fails to (showing that series do not end)?
More generally speaking, can we fully conceive of the mark of arising without 'no arising', and / or vice versa?
No, this came across pretty clearly. I think your question perfectly captures the essence of the debate that happened several centuries after Buddha's death.
Back then there was a lot of work done by Sarvastivada followers and other early schools trying to enumerate the dharmas as one final exhaustive list, sort of like the table of chemical elements these days.
As they were doing that, one dharma that kept standing out was "nirodha", defined as "not arising". This was supposed to be referring to Nirvana/Nibbana, not-arising of the chain of dependent origination (and suffering) due to the absence of passion and grasping, due to the absence of wrong views, due to the absence of ignorance. (Some said nirodha was unconditioned/asamskrta because it owed its existence to absence of necessary conditions, rather than presence - a philosophical sleight of hand.
While I would not personally equate nirodha with nirvana and idolize all kinds of not-arising, the point that Nirvana is effected by liberation from wrong view -- and from the "right view with asava" -- does make it look like a kind of nirodha. That said, the unconditionality of Nirvana comes from it being immune to all imputations that serve as the bases of conditions, so Nirvana is seen through nirodha of imputations, Nirvana is not nirodha itself. In other words Nirvana is tathata, the unity of Two Truths, while nirodha of Ignorance is just a way to see the Absolute.)
The same questions as the ones you asked, whether not-arising was a thing, and can it be said to truly exist other than relative to arising, were undoubtedly brought up and led to a debate about the notion of existence and not existence, and the concept of svabhava (existing by itself or from its own side). Well, it wasn't just about this dharma, there were other controversial dharmas like akasa (space, defined as "no resistance to movement"), and even simpler dharmas like heaviness, but non-arising was likely at the top of the list.
This debate, in turn, is what (probably) led to the emergence of the notion of Emptiness and the tradition of Madhyamaka as we know it today.
The exact details of that development are still not clear to me, i.e. whether it was directly inspired by the prajnaparamita movement or just happened to come to the same conclusion from another side, but the point that Madhyamaka was born in or close to Gandhara from polemics around Sarvastivada Abhidharma is pretty much settled for me.
To answer your question directly: the Buddhism split at this very question. Mahayana says all dharmas (~"things") are relative and are delineated approximately by an observer. Mahayana says the very notion of "existence" or "being a thing" is, too, an approximated delineation. According to this view, arguing about something being or not being really real is foolish and betrays fundamental confusion on one's part about the nature of reality and the Buddha's teaching. The non-Mahayana (so-called "Hinayana") schools stand by their conviction that some dharmas are really real and not mere relative delineations/imputations - though they don't all agree on the exact list, especially when it comes to the asamskrta/unconditioned dharmas. But yeah, at least in some of the "Hinayana" schools nirodha (~not-arising) is considered a real dharma, something that actually exists.
Consider a pool of water. If there is an 'arising' — something bubbling up from the depths — the surface of the pool ripples and glimmers in the light. If there is no 'arising' the surface of the pool is still and calm.
It's the nature of the pool to be still and calm. An arising is a disturbance.
The mistake all of us make (early on) is to presume that the pool is identical to the ripples and glimmers that we see on the surface. We see the effects of arisings and convince ourselves that arisings are the essence of the thing on which we see the effects. We cannot see the pool for the ripples.
'No arising' isn't a thing. It's the absence of a thing that the mind expects: the proverbial waiting for the other shoe to drop.
Two fail in liberation from suffering, not arising of it: those after becoming/being/arising, and those after not-becoming, not-being, not-arising, good householder.
