Don't these experiments suggest that locality has to be abandoned in the quantum realm? First a quick recap because maybe my understanding/assumptions are flawed (you can jump to the question highlighted in bold below if you like to):
Bell's inequalities require that an accurate(ly predictive) theory cannot include local realism (or have local hidden variables).

Bell's theorem, combined with experimental testing, restricts the kinds of properties a quantum theory can have, the primary implication being that quantum mechanics cannot satisfy both the principle of locality and counterfactual definiteness.
  ~WP:Interpretations of quantum mechanics

In quantum entanglement a quantum system's particles' states are correlated in a way that makes it appear as if measurement of one particle instantaneously influences the other independent of the space between them. Einstein couldn't believe that a theory that includes such a seemingly nonlocal "effect" could be true and complete and called it "spooky action at a distance". In 2015 two loophole-free (with the exception of at least superdeterminism) experiments confirmed Bell's theorem even better and proved instantaneous nonlocal collapse of a particle's wavefunction. In 2018 two experiments closed two remaining loopholes and in 2019 Bell's entanglement was photographed.
However, it seems like many - or most involved scientists - believe that there's nothing roughly describable as "action" involved: according to them and the preferred theories the measurement of the particle does not affect the remote particle nor anything else that in turn "affects" the remote particle. Neither, conversely, does the universe or any other mechanism have the measurement have an instantaneous "affect" on the remote particle. That's because according to the principle of locality two spatially separated objects cannot have instantaneous mutual influence - within our universe no effects can be mediated faster than light. Hence and because the 2015 experiment only disproved realist-localist theories and not locality for example this article on the New York Times was wrong and remains uncorrected - it said [emphasis mine]:

[...] experiment that they say proved one of the most fundamental claims of quantum theory — that objects separated by great distance can instantaneously affect each other’s behavior.
  The finding is another blow to one of the bedrock principles of standard physics known as “locality,” which states that an object is directly influenced only by its immediate surroundings

At the same time it looks like the seemingly instantaneous "effect" also isn't due to "hidden variables" which would make it like "Bertlmann's socks" (if you know you have a sock of color y you know Bertlmann must have taken sock of color x).
But does that mean the wavefunction couldn't get altered at entanglement? Wouldn't an altered wavefunction or something similar allow for random and unpredictable but correlated results? Results don't have to be predictable or non-"random" to be correlated. That's probably a separate question though. But it seems like the wavefunction could also be considered a (non-hidden) variable: here @FrédéricGrosshans writes:

For quantum theory, the wavefunction of the object is the hidden variable.

And is the collapse of the wavefunction better described as a change of our knowledge than a "real" (beyond the realness of our local knowledge) "change" of the remote particle? Here @LubošMotl apparently popularly claims just that:

But this step, in which the original overall probabilities for the second particle were replaced by the conditional probabilities that take the known outcome involving the first particle into account, is just a change of our knowledge - not a remote influence of one particle on the other.


A lot of people everywhere are saying that basically Bell's inequalities mean that one has to abandon locality or realism.
The same goes for the conception that realism is to be abandoned and locality to be preserved.
Here @pwf explains why traditional theories which preserve locality are more common:

scientists have settled on the unrealistic theory (i.e. QM), it's because, compared with the only viable realistic theory (so far), which is Bohmian mechanics, it was developed first, it's easier to use for computation, and it has proven highly useful and successful

and @Timaeus in simplified terms explained another reason for the more common standpoint, saying:

What is the point of a realistic nonlocal theory? If someone far away can mess with your realist stuff then how can you make any local predictions unless you average away the influence far away people can have on your stuff.

Furthermore here the top answer says:

Locality is the assumption that an object can be influenced only by its immediate surroundings by the events which took place in its immediate past. All classical and quantum field theories depends on this assumption in an essential way. Non locality implies that two events which are separated from each other by space-like separation can affect each other. Some people demand (imho) falsely that EPR type entanglement violates locality. In reality in never does. All one need to abandon is realism. Entanglement just shows that there exists quantum correlations between particles which were in past had some common origin. It also shows that if it were a classical world then the EPR entanglements effects were nonlocal. But we live in a quantum world and there is no non locality.

In short as @EmilioPisanty put it:

Modern phycisists, roughly speaking, tend to throw away realism, to keep the locality.

And here authors write:

Measuring or otherwise interacting with a quantum system S has no effect on distant systems from which S is dynamically isolated, even if they are entangled with S.


Now a number of experiments confuse me as they seem to suggest that locality should be abandoned and that the "spooky action at a distance" might truly be some form of non-local "action" instead of basically "(un)spooky correlation at a distance":


*

*"Experimental nonlocal and surreal Bohmian trajectories"
which was reported with:

The team thinks this means that the trajectory of the first photon changed the probe’s polarisation – in line with Bohm’s ideas on non-local interactions.

and

In the most recent experiment, Steinberg and colleagues showed that the surrealism was a consequence of non-locality -- the fact that the particles were able to influence one another instantaneously at a distance. In fact, the "incorrect" predictions of trajectories by the entangled photon were actually a consequence of where in their course the entangled particles were measured. Considering both particles together, the measurements made sense and were consistent with real trajectories.
  Steinberg points out that both the standard interpretation of quantum mechanics and the De Broglie-Bohm interpretation are consistent with experimental evidence, and are mathematically equivalent.

and in the study it says:

we demonstrate the nonlocality present in Bohmian mechanics by showing that the trajectory of photon 1 is affected by the remote choice of how to measure photon 2


*"Experimental demonstration of a quantum shutter closing two slits simultaneously" (and "Nonlocal Position Changes of a Photon Revealed by Quantum Routers"):

can a single shutter simultaneously close two slits by effectively being in a superposition of different locations? [...]
  This experimental demonstration, where the strong measurement and non-local superposition seem co-existing

and the follow-up study is reported with:

The apparent vanishing of particles in one place at one time—and their reappearance in other times and places—suggests a new and extraordinary vision of the underlying processes involved in the nonlocal existence of quantum particles.

and

“it’s possible a superposition is a collection of states that are even crazier,” Elitzur says. “Quantum mechanics just tells you about their average.” Post-selection then allows one to isolate and inspect just some of those states at greater resolution, he suggests. Such an interpretation of quantum behavior would be, he says, “revolutionary”—because it would entail a hitherto unguessed menagerie of real (but very odd) states underlying counterintuitive quantum phenomena.

but also

“The experiment is bound to work,” says Wharton—but he adds it “won’t convince anyone of anything, since the results are predicted by standard quantum mechanics.”
  [...]
  Elitzur agrees their experiment could have been conceived using the conventional view of quantum mechanics that prevailed decades ago—but it never was. [...] And if someone thinks they can formulate a different picture of “what is really going on” in this experiment using standard quantum mechanics, he adds, “well, let them go ahead!”


*"Entanglement Swapping between Photons that have Never Coexisted" (and "Experimental delayed-choice entanglement swapping"):  


The observed two-photon state demonstrates that entanglement can be shared between timelike separated quantum system 

What I don't understand then is why people are saying that one has to choose between giving up locality or realism (/counterfactual definiteness/non-contextuality) with the latter being preferred by most: shouldn't locality at the quantum level (also) be abandoned due to these experiments? Or are they just proving "quantum nonlocality" with that only referring to the seemingly nonlocal effects of entanglement which only appear nonlocal but most likely aren't?
If locality is to be abandoned wouldn't the terminology "action at a distance" be correct to/with an arguable degree/certainty and due to the lack of a better word?
Note that recently many also add freedom (of choice/non-conspiracy/non-determinism) as a third variable. Also regarding the first experiment I'm not sure whether those trajectories are only relevant to the Bohmian interpretation and whether the experiment is irrelevant to non-locality in standard QM (why would it?)?
(And isn't it the case that violation of locality doesn't imply violation of locality on the macroscopic level and relativistic causality? For example in here it is written that "causality only imposes a subset of no-signaling conditions" and that "relativistic causality, if strictly applied, gives much more freedom for correlations than the no-signaling conditions. The latter were just inherited from quantum mechanics and certainly make sense if the only carriers of physical interactions are local fields that manifest themselves as particles". They also write that "not only quantum theory but any future theory that might contain the quantum theory as an approximation is now expected to incorporate nonlocality as an essential intrinsic feature" - wouldn't the "nonlocality" of quantum mechanics mean that locality has to be abandoned?)

There are a number of possible resolutions to this that I can think of:


*

*There are many different kinds (and degrees/..?) of locality (and different situations where those are relevant) and those experiments only disprove some of them / when quantum mechanics speak of (non-)locality in QM they're referring to something else (depending on the context) and only use the same word

*I (and others?) misinterpreted the experiments

*There were flaws or shortcomings in the experiments: for example aren't the the entangled particles in experiments in close proximity and aren't the results just statistical instead of literally two photons only (highly segregated at least) and couldn't there be limiting technical imprecisions (or physical ones like the uncertainty principle) that could invalidate relevant conclusions from the experiments?

*Something else

 A: Sorry, the answer is rather lame, and has more to do with PR, impact factors, and science journalism than actual physics.
The point is that "locality" comes with many different meanings. The straightforward meaning is that a theory is local if you can't violate relativistic causality, i.e. you can't get information about anything outside your past light cone. The "problem" is that it's boring to talk about this, because everybody reasonable (including everybody you cited) agrees that QM is local in this sense.
So instead we usually talk about a different notion of locality. Instead of focusing on things that can actually be observed, we say a system is nonlocal if any part of our description of the system in the particular formalism we're using at the moment can be affected faster than light. Of course this means that locality depends entirely on the formalism you're using.
To take a possibly comically simple example, consider the following two models for coin flipping.


*

*Every coin flip is heads or tails, with a 50/50 chance.

*Every time you flip a coin, that coin instantly figures out the last result of any coin flip anywhere in the universe, faster than light. Then, with a 50/50 chance, it either gives the same result or the opposite result.


Both theories are local in the first sense -- you can't use the coin flips to signal anything. However, (2) is nonlocal in the second sense, because in order for us to compute what a coin is going to do, we need nonlocal information. These two "interpretations" of coin flipping are completely observationally equivalent in every way, so in some sense, they're equally good scientific theories for coin flipping, in the same sense that different interpretations of QM are equally good. 
The difference between the coin theories, just like the difference between interpretations of QM, is that if you use (2) you have to do uglier computations and think much harder. For instance, if you added enough bells and whistles to it, it might no longer be obvious that it's local in the first sense. 
All of these sensational news articles you see from trendy optics papers boil down to choosing a description of quantum mechanics that's more like (2) than (1). Outside of this tiny subfield of Nature/Science bait, nobody actually uses descriptions like (2) for anything. That's because the main benefit of using descriptions like (2) in optics is that it sounds crazier, allowing one to make precisely the kinds of statements that confused you above, go viral, and get one's paper accepted.
A: Locality does not have to be abandoned.
But: 
QM shows that some experimental correlations have no causal explanations. What the experimental confirmations of the violation of Bell's inequalities show is that the random part of QM, the Born rule, is to be taken seriously: no deterministic mechanism is hidden beyond [what would be seemingly] random results, because such a mechanism would need hidden variables to maintain the spacelike separated measurement correlations, and hidden variables are precisely what Bell's result allows to discard.
Now does the observed existence of non-causal correlations imply that locality is invalid? As knzhou explains in his answer, no, because the truly random nature of the measurement outcomes does not allow action-at-distance, and action-at-distance would be the true marker of non-locality. 
It is because action-at-distance is still not possible, that I first said that QM shows non-causal correlations: these are two sides of the same coin. Causality and locality are intimately connected, by the very structure of spacetime.
So the situation we have is, how to make sense of non-causal correlations? Again as knzhou explains, in a sense we may not need to deal with this question at all. After all, no physics will be changed by any insight we may get about it, because we will still not be able to get any action-at-distance from that insight. But it is still a tremendously fascinating question, and answering it correctly may allow us to advance our physical understanding of nature to the next step. 
A: I can only answer with regard to the first experiment that you mentioned 'Experimental nonlocal and surreal Bohmian trajectories' as I do not understand the others sufficiently, but hopefully this will help. In your question you write 'regarding the first experiment I'm not sure whether those trajectories are only relevant to the Bohmian interpretation and whether the experiment is irrelevant to non-locality in standard QM'. I'm pretty sure that this suspicion that you had is correct, but in order to explain this we need to examine what Bohmian mechanics involves.
Bohmian mechanics is an interpretation of QM produces exactly the same experimental predictions as standard QM, but it explains them in a different way to other interpretations. As you noted, Bell's theorem tells us that we need to throw away locality or realism and most interpretations decide to throw away realism. Bohmian mechanics  throws away locality and keeps realism. It states that as we fire particles at a double slit, each one takes a well-defined trajectory from the slits to screen - this is the realism that is not present in interpretations such as the Copenhagen interpretation. However, the potential which guides the particle along this trajectory comes from instantanteous interaction with other particles over arbitrary distances -  this is nonlocal. Thus, by being realist, Bohmian mechanics must discard locality. It is important to note that the predictions made by Bohmian mechanics are, by design, identical to the predictions made by standard QM.
In the paper you mention, the authors are addressing a point (which admittedly I don't understand) about the Bohmian trajectories. They claim that there are certain properties of these trajectories which other people have misunderstood, and which are better understood once you take into account the fact that Bohmian mechanics is nonlocal. This is probably an interesting result, but doesn't mean that we have to abandon locality in all quantum physics, we just need to abandon it when we have realism, as in Bohmian mechanics. In other words, Bell's theorem still holds.
Hopefully someone else can address the other two papers for you - I'm also interested to hear what they have to say!
