Do photons generate gravitational waves since they affect with their energy the stress tensor? The gravitational waves are fact. They are produced in a way predicted 100 years before by Einstein. Anything with energy affecting stress tensor of space time produces them. What does it happen with moving photon losing its energy in a way creating gravitational waves? Eventually it will dissapear. So finally when all matter decays and becomes in form of energy respectivly photons and they decay becoming gravitational waves in the end we will finish with vibrating spacetime? Is that the case?
 A: Right, so photons will not generate gravitational waves on their own; it requires an acceleration of mass and furthermore an asymmetry in how it's accelerating (a radially vibrating spherical shell, for example, is not viable as a gravitational "speaker" for transmitting gravitational waves to the cosmos, no matter how massive it is).
However, it is right now plausible that due to a combination of the accelerating expansion of space and black hole evaporation, maybe every particle in the universe will someday be housed off into its "own pocket of spacetime," within which there are no other particles. One very interesting idea about cosmology, called Penrose's "conformal cyclic universe", says "well, maybe the only reason any particle experiences time is due to its interactions with other particles giving it an effective mass, which somehow acts as an effective clock of frequency $f = mc^2/h.$ But if particles can't interact with other particles then they might lose this mass and travel off to infinity at the speed of light. Well if every particle is massless then nothing in the universe experiences time anymore, and we can just continue the universe to $t=\infty$ because it's not like there are any interactions anymore which care." Apparently if you take this idea really seriously, you may have that the $t=\infty$ state looks a lot like a $t=0$ big-bang singularity when seen from a different perspective: so maybe there actually is a cyclic universe even with an accelerating expansion of space: but maybe the 'cycles' require looking at the universe from a "God's eye view" where you can look out past the $t=\infty$ of these coordinates that our material bodies are helplessly stuck in.
A: Use of the word "photon" is where quantum mechanics becomes involved and is where I'm unqualified to answer! 
However, classically, yes, the electromagnetic field has some stress-energy and so contributes to gravity. And yes, if two absurdly energetic "lumps" of light pass each other, you will get a gravitational wave same as two ultrarelativistic lumps o matter passing each other. However, this does NOT imply that all light will decay to gravitational waves. It doesn't even imply that all matter will "stop moving" or become at rest with respect to all other masses.
In fact, the equations of general relativity have been solved in the case of uniform radiation in every direction. This is the radiation dominated era  and I am referring to the Friedmann equations. This doesn't represent the future of our universe well, though, because our universe isn't going to be radiation dominated in the future, it will be dominated by dark energy! In this era, uniform radiation will die out to another reasion: As space expands, any wavelength of light will stretch out, getting redder, and redder, and redder, until it's of vanishingly small energy density. This is almost exactly what happened to the light from the big bang. If you look at the black night sky, looking past all the stars/dust/nebulae, you'll end up looking at the hot remnants of the big bang. The expansion of spacetime stretched the short high-energy wavelengths into long wavelengths, killing the energy density of the light. But that's different than emitting gravitational waves.
