What probability distribution the detection counts have? As far as I know in quantum mechanics each particle have a separate normalized wave function that can be used to calculate that the particle can be found somewhere. Or more practically to determine the chance that the particle is detected by a perfect detector. Each particle have the same chance.
Now let's assume we have a photon source that allows us to emit a set amount of photons (is it possible?).
Then if each photon have a separate wave function then the detection counts:


*

*Never exceed the number of photons emitted.

*Should have binomial distribution.


But in practice regarding the shot noise we saw that the detection counts have Poisson distribution. Poisson distribution arises when we test for an improbable event lot of times, see Poisson limit theorem for proof. 
This would mean lot of things:


*

*Lot of particles are emitted but we have a horribly bad detection efficiency.

*Paricles doesn't have separate wavefunctions for each of them.

*You cannot make a fixed number photon source so the question is pointless.

*There are no free photons, they are just units of the energy exchange of for that given frequency, and that exchange happens randomly when the material is subjected to electromagnetic radiation (my pet theory, sorry).


The big question is: have we observed the binomial distribution with reasonable confidence? 
 A: Quantum mechanics is not about particles but about quanta. The quanta are the quantized changes of a single  object called a quantum field. One can not, in all generality, assume that single particles have "independent" wave functions. That's ca useful approximation some systems, but it is certainly not the case for systems that emit photons. Instead we have to take spatial and temporal coherence into account and this is especially true for systems that emit a fixed number of photons. 
On the other hand, if we don't want any correlation between photons, whatsoever, then we have to let go of the fixed particle number requirement and go with a thermal photon source, which acts like a large number of random emitters. In that case, however, only the average flux is fixed. 
Beyond that I don't understand your question. Do we understand photon statistics of photon sources and detectors. Yes. Is it binomial? No.  
A: Your main assumption is that "we have a photon source that allows us to emit a set amount of photons," and you ask if that's possible. Certainly. However, except for when you are calibrating an emission source, you are usually just measuring a source that is not under your control. Then, for sheer probabilistic reasons, event counts should follow a Poisson distribution. This isn't even necessarily a quantum effect, since the randomness could be just as easily due to classical thermodynamics.
