About 'de Broglie hypothesis' and the double slit experiment EDIT: As i mentioned in my original question, i do not have the background to fully understand @Timaeus answer (which was very detailed indeed). 
I would appreciate if someone could give a more 'classical physics'  answer ,even not so detailed ,in order to clarify some things my self a little bit more. 
In addition i would like to know  the difference between the term 'wave' and 'wavefunction' and how two uncharged particles would interfere in the experiment demonstrated by a single's particle's emission source.
Without enough theoretical background in physics , I post this question which actually has two related parts.
Wavefunction and 'de Broglie hypothesis':
As far as I can understand the wavefunction of massless particles, is described by the magnitude's change of a certain particle's property, i.e. the wavefunction of a photon is described as the changing of the intensity of it's EM field  over time. This wave moves through space with velocity $C$ and carries energy equal to $hf$.
On the other hand 'de Broglie hypothesis' suggests that 'all matter has wave properties' and the wavelength of this function is equal to $\lambda=h/p=h/mv\cdot \gamma^{-1}$. 
Now here comes my first question: In which particle's property is this wavefunction related to?
 Or does this wavefunction actually describes the particle's motion (~)  with it's simultaneous transport through
space with velocity $v$,carrying $\rm{KE}\;?$
b) In the double slit experiment that is demonstrated by a  single's electrons emission source, what physical property is the interference pattern related to ?
 Is this pattern the result of the interference of accelerating electron's EM wave or something else?
 If the source emits 'n' single electrons, how many arrivals of matter do we detect on the screen?
 A: There are two approaches to quantum mechanics: nonrelativistic quantum mechanics, and quantum field theory.
In quantum field theory, there is one wave in physical space for each type of particle/antiparticle. A photon field, an electron/positron field, a muon/antimuon field, and so on. But the fields are operator valued, and quite complex. And there isn't a clean thing that corresponds to a single particle. The one field collectively represents all the photons and such in the whole universe.
In nonrelativistic quantum mechanics you have to pick a number of particles, such as $n$ particles and then each has a spin space $\mathbb C,$ or $\mathbb C^2,$ or $\mathbb C^3$ or $\mathbb C^k$ and then the wavefunction is a function from $\mathbb R^{3n}$ (note this is configuration space, which is much larger than physical space) and it goes into $\mathbb C^{k_1}\otimes\mathbb C^{k_2}\otimes\dots\otimes \mathbb C^{k_n}$.

This wave moves through space with velocity $C$ and carries energy equal to $hf$.

That never happens.

On the other hand 'de Broglie hypothesis' suggests that 'all matter has wave properties' and the wavelength of this function is equal to $\lambda=h/p=h/mv\cdot \gamma^{-1}$. 

The $p$ is canonical momentum, not mechanical momentum. And even if it were mechanical momentum, the correct formulas are things like $\vec p=E\vec v/c^2$ (which holds for all particles) not $\vec p=\gamma m\vec v$ (which only holds for massive particles).
Just because some formulas are more popular and they hold for some special cases doesn't make them the correct formulas.

Now here comes my first question: In which particle's property is this wavefunction related to?

The wavefunction describes all properties of all particles.

Or does this wavefunction actually describes the particle's motion (~)  with it's simultaneous transport through
  space with velocity $v$,carrying $\rm{KE}\;?$

It's a wave in configuration space. So the probability current describes flows in configuration space. A point in configuration space is an assignment of locations to all particles. So it's a flow from a configuration of all particles to another configuration of all particles.

b) In the double slit experiment that is demonstrated by a  single's electrons emission source, what physical property is the interference pattern related to ?

The interference pattern is in the residuals of the locations of the particles.

Is this pattern the result of the interference of accelerating electron's EM wave or something else?

Something else. The interference of the spin in configuration space. With a residual over the locations of the screen. As measured by the frequency of a statistical ensemble which is related to the overall amplitude of a single instance of the ensemble.

If the source emits 'n' single electrons, how many arrivals of matter do we detect on the screen?

Less than $n$ if some hit the barrier on the way to the screen. The ensemble is all $n$ and in nonrelativistic quantum mechanics you get the prediction of the frequency of different locations from the wavefunction for just one electron.
