Why does transition from one electron shell to another shell always produce massless photon? When electrons transition from a higher energy state to a lower energy state (energy difference $E$), they produce massless photon with frequency $\nu$ where 
$ \Delta E= h \nu$
(h is Planck constant). We know energy-mas relation $ E=mc^2$. Why not create  some kind particle, in this case a particle that has mass m that we could calculate from the energy difference of the two states of the electron? Is there any kind critical energy difference $\Delta E_c$ such that lower than $\Delta E_c$ always  is creating photon and higher than $\Delta E_c$ its  value create particle with mass?
 A: You are asking about when a electron transitions from a higher energy level to a lower energy level so I will assume you are asking about relaxation. There are other type of transitions, like the Auger effect.
Now when a atom/electron relaxes, it moves from a higher energy level to a lower energy level as per QM, as you state, and releases a photon.
You are asking why this transition releases a massless gauge boson.
Now the photon is an elementary particle, part of the SM, massless, pointlike. Photons always travel at speed c in vacuum, when measured locally.
You are asking why this gauge boson is massless.
There are two ways to look at this:


*

*gauge invariance, if you want to describe a theory with a zero mass vector and relativity, you have to have gauge invariance. And the photon is massless because it is the mediator of the EM force which is long range. It is because of the unbroken U(1) gauge invariance of the EM force.


Though, the gauge fields may become massive via Higgs (W,Z bosons). But that is a short range force.
Why can't gauge bosons have mass?
How does gauge invariance prevent the photon from acquiring a mass?


*As per SR, anything that travels at the speed of light, cannot have rest mass. This is because it would cost an infinite amount of energy to speed up a massive particle to speed c.


https://en.wikipedia.org/wiki/Photon
https://en.wikipedia.org/wiki/Special_relativity
A: There are a few reasons why the particle produced needs to be a photon. Aside from conserving energy, we also need to conserve momentum, charge and spin, for example. So you would need to ask what other particle, instead of a photon, could be emitted while satisfying all those conservation requirements. 
If you just consider energy and spin conservation, the total amount of energy available in electron transitions in an atom is small, and not enough to make any of the other massive Bosons. To use your terminology, the maximum energy difference in electron transitions, Δ, is way below the energy Δ you would need to create any of the other known massive particles that satisfy the other conservation requirements.
A: It's just a conservation of energy equation.
An example of such a case when an electron jumps to a lower shell it emits a photon and this photon itself is captured by an electron in the outer shell and that electron gets emitted from the atom. So in principle till all the conservation laws are satisfied there is always a finite but very less probability for it to happen than just a emission of photon when an electron jumps to a lower shell.
