Does the diffracted electron radiate photons? When electron is diffracted after the slit it might follow different direction, than before the slit. That means, that going through the slit it gains some acceleration. And accelerated charge emits photons.

Thus - does the diffracted electron radiate photons?
 A: I believe the answers given so far are off the mark and do not directly answer the OP's question: do diffracted electrons emit bremsstrahlung radiation/soft photons?
So I give the answer here:  The answer is a resounding yes!  
However since the diffracted electrons are scattered in the far forward direction (tiny scattering angles), the momentum/energy transfer is minute and therefore the emitted photons have extremely long wavelengths.  Therefore, in any practical diffractive experiment, the radiation is undetected.  That the radiation is not detected is very important in order to maintain coherence of the system, and therefore leave the diffraction pattern unspoiled.
(I think CuriousOne was alluding to this kind of answer in the comments section of others' answers)
An amusing thought just crossed my mind: it may be possible to observe the radiation for components of the electron trajectories deflected at sufficiently large scattering angles, in which case it is possible to deduce, in principle, the direction of the scattered electron, leading to the loss of the interference pattern at larger angles!
A: 
Thus - does the diffracted electron radiate photons?

In your question, you use terms acceleration from classical mechanics and photon from quantum theory. Since these theories are not mutually compatible, the question is badly stated. To get a useful and clear answer, you have to state which theory you are asking.
If you ask "does the electron radiate EM waves when it passes through the slit?" it is a question in the realm of classical theories and the answer is yes based on Maxwell's equations, because electron accelerates in the vicinity of the slit and any accelerated charged particle radiates EM waves.
On the other hand, if you ask "does the electron radiate photons when the double slit experiment is done?" the answer is not so clear, because it is a question in the realm of theory with photons, e.g. quantum field theory. The answer depends also on other assumptions such as initial conditions for the electron and EM field. The corresponding description of diffraction in time would be quite complicated, but no doubt the quantity representing EM field will be non-trivially time-dependent, especially near places electron is present during the diffraction. So I would say there is radiation even in quantum field theoretical description.
I would not insist this means photons are produced, because the field may not even be in state that could be described by the concept of photon number. When people talk about photon transition, it is usually an expression of the idea that some measurement suddenly changed the state of the EM field into state with definite photon number. That is not necessary to calculate solutions of the equations, and describe the double-slit experiment.
A: As in my reply to the question Diffracted electron - where does it gain additional momentum? it depends on the size of the slit and the energy of the electron. If the energy of the electron is large enough and the slit large enough, the scattering on the field of the atoms at the sides of the slit will radiate a photon and change the direction of the electron.
If the size of the slit and the energy of the electron are of the order of h_bar where the wave nature of the diffracted electron will become evident, the answer to this is the same as the answer of "why the electron does not fall on the nucleus but stays about the atom without radiating photons". It led to the Bohr model of the atom to start with and then to the full Schrodinger equation and solutions of the hydrogen atom. These solutions give probability distributions for locating the electron.
Thus the answer is that the electron is a quantum mechanical entity, and the solution of "electron of vector momentum p + slit" has as a solution a probability distribution that allows the electron various paths, without radiating any energy. This is related to the Heisenberg uncertainty principle.
Here is the double slit experiment with accumulation of single electrons:

>Electron buildup over time
One observes that even though a single electron can be anyplace on the screen, the probability distribution built up during the experiment displays an interference pattern. It is a quantum mechanical effect dependent on the geometry of the slits but it shows that the electron is not radiating away random photons, as there would be no interference pattern. The question is about a single slit, but the physics is the same, just the boundary conditions are different for the quantum mechanical problem.
Edit after comments, as the answer has gotten  a number of negative votes, to clarify radiation or not by the electron
Take the electron at the maximum off center location in the top picture. It is evident that it has gotten a momentum in the x direction. Treated classically, how could it get there with all its energy? Classically a neutral ball can scatter elastically, momentum being conserved by the whole solid where the ball would hit/graze. Classically though a charged particle changing direction drastically should radiate and it would lose energy, more energy the higher the angle. This random loss of energy would destroy an interference pattern, as the questioner guesses. As interference is observed and we know that the framework is quantum mechanical at the level of atoms and electrons, a wavefunction must exist giving the probability distribution of finding an electron given the boundary conditions of two slits with no energy loss , similar to the energy levels in atoms where the electron might be in any position in the orbitals without radiating.
