Why do we only detect Minkowski particles? I'm having trouble building intuition of the Unruh effect, altough I can follow the proof using Bogoliubov transformations.
When studying a massless scalar field in 1+1D QFT, we start from the classical Klein-Gordon equation. We decompose solutions as left and right moving ones (something like $f(x+ct)$ and $g(x-ct)$). We then do a Fourier expansion of both solutions, so they are a sum of Minkowski normal modes: $\exp(-i\omega(x±vt))$. To quantize the theory, we simply promote the Fourier coefficients to creation/annihilation operators, which create and destroy particles.
We could choose another expansion in terms of other modes, $\exp(-i\omega h(x±vt))$ where $h$ is a new function, so a vacuum associated to these modes would contain particles according to Minkowski normal modes.
I don't understand then, what expansion should we choose. I've read that we should choose an expansion in terms of a null coordinate that describes the observer, so positive and negative frequencies don't mix.
Is this choice unique?
To give an example, if we accelerate, we would see photons because the Rindler vacuum is different from the Minkowski vacuum. Does this mean that we could make photomultipliers that don't detect photons when accelerated, but will start clicking at rest?
It seems that a justification that we detect Minkowski particles is that all inertial observers agree on them, but I don't see how this is relevant.
 A: On why the Unruh effect happens
I believe I should start this answer by commenting on this:

if we accelerate, we would see photons because the Rindler vacuum is different from the Minkowski vacuum

That is not the reason an accelerated observer sees photons. The state of the quantum field does not depend on coordinates or reference frames. All observers agree on what is the field's state. This becomes particularly clear when one attempts to formulate the theory in the algebraic approach (as advocated in Wald's book).
The reason an accelerated observer sees particles in the exact same quantum state an inertial observer sees none is because they have a different notion of what a particle is. This is similar to how the twins in the twin paradox measure different time intervals between departing and finding each other again: they have different notions of time. In spite of that, they both agree on physically significant matters (such as the quantum state of the field, or the causal order of timelike related events).
That being said, let's move on to the actual questions.
On the choice of null coordinate
This question has to do with formulating Quantum Field Theory in Curved Spacetime in a particle-like language, which employs this null coordinate construction. While this is a convenient approach in some situations, it is worth reminding that it is by no means necessary or fundamental. One can also formulate the QFT without these sorts of choices by employing the algebraic approach, which gets rid of any ambiguities.
However, if one happens to be interested in what is measured by an observer whose worldline follows the orbit of a timelike Killing field, it is possible to do this construction. The null coordinate will be determined by this Killing field and it is indeed unique. I particularly recommend Wald's book in this, as (at least in my opinion) the construction done in there leaves no doubt that the Killing field is sufficient to determine the Fock space you're desiring. The intuitive idea is that the Killing field determines the notion of energy you're working with, so you are now capable of distinguishing which modes have positive energy and which have negative energy.
On photomultipliers

Does this mean that we could make photomultipliers that don't detect photons when accelerated, but will start clicking at rest?

Things are a bit more complicated. The Rindler vacuum, which is the quantum state in which an accelerated observer sees no particles, is singular at the Rindler horizons, meaning it is not a physically acceptable state in Minkowski spacetime. If somehow you add matter to the spacetime in a way that changes the metric in the region of the Rindler horizon, then sure, it could be that an inertial observer would detect photons (does depends on details and would need a computation to be fully sure of a specific scenario).
Why we detect Minkowski particles

It seems that a justification that we detect Minkowski particles is that all inertial observers agree on them, but I don't see how this is relevant.

We detect Minkowski particles because we move in inertial trajectories. Minkowski particles correspond to modes such that $- p_a t^a > 0$, where $p^a$ is the four-momentum of the mode and $t^a$ is the Killing field associated with Poincaré time-translations. If turns out that inertial observers are precisely those that measure energy according to $- p_a t^a$.
It should be recalled, however, that all observers will agree on physically real predictions. For example, suppose the quantum field is prepared in the Minkowski vacuum. An accelerated observer departs with a particle detector coupled to a bomb: if the detector detects a particle, it will explode the bomb. Now, both observers should agree on whether the bomb exploded or not. And they do. They just disagree on why it went off.
For the accelerated observer, the detector simply absorbed a Rindler particle. For the inertial observer, the detector emitted an inertial particle and tunelled to the excited state as a radiaction reaction to that emission (as described in detail in Unruh, W. G. and Wald, R. M. (1984), "What happens when an accelerating observer detects a Rindler particle", Phys. Rev. D 29, 1047). Hence, they disagree on the mechanism, but agree on the physical consequences. Just like the twins disagree on elapsed time, or how different inertial observers might attribute the interaction between a charge and a current-carrying wire to electric and/or magnetic forces, depending on the reference frame.
In Short
We measure Minkowski particles because we are, to a good approximation, moving inertially through spacetime. However, it should be noted that an accelerated observer would agree with us on what is physically and there is no inconsistency in different observers measuring different notions of particles. Also, the Unruh effect is due to different observers having different notions of particles, not due to them observing different quantum states.
