What is meant by rest in rest-mass? As far as I know only photons are considered to have no rest-mass. In common words when it doesn't move it 'disappears'. 


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*Electrons and quarks should have a rest-mass. But are they really at rest? 


In atoms and molecules is always a kind of zero-point energy left which implies that there is still some  'movement' in the particles. 


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*So when an electron or quark is really at rest does it still exist? 

*Or does it 'disappear' just like a photon? 

*And hence is there really a difference in mass-property of particles or is the equation $E=mc^2$ already suggesting that there is no real property difference in mass between a photon and electron? Or there is?
 A: The term rest mass is a poor one because it implies it's the mass measured in the rest frame. But photons have no rest frame, and indeed any particle subject to some form of confinement has a $\Delta p\gt 0$ so its rest frame is somewhat poorly defined.
The modern term is invariant mass, which is simply the mass in the equation for the total energy:
$$ E^2 = p^2c^2 + m^2c^4 $$
Using this definition the electron has an invariant mass of $\approx 511$keV while the photon has an invariant mass of zero.
While, as you suggest, it can be difficult to get an electron into a state where it is completely at rest you can approach this state to any desired degree of accuracy. And if you do then you'll find the rest mass approaches the invariant mass.
A: Rest mass means the mass which would appear if a paricle were at rest. Do not confuse between particles and photons. These particles are metarialistic particles behaving as energy in some circumstances. Being matter they possess rest mass. While photon is a bundle of energy behaving sometimes as a particle and hence can not possess a rest mass or more precisely, invariant mass.
A: The main issue in your question is that it assumes that motion is absolute. Which it is not.

As far as I know only photons are considered to have no rest-mass. In
  common words when it doesn't move it 'disappears'.

Wrong conclusion. It just always move. There is no reference frame where a photon does not move at speed $c$. 

Electrons and quarks should have a rest-mass. But are they really at
  rest?

There always exists a frame of reference where any object which has a rest-mass does not move. In other words, if you move along the object, it doesn't move for you. But you can't do that with a photon, because you cannot move at speed c (because you have a rest-mass).
Now "objects" such as quarks and electrons are not classical at all, so most of the time they just have no specific position nor momentum, no trajectory. In that case it doesn't make much sense to wonder how they move; when they are bounded (which for quarks is always the case) it is better to talk about their energy. Then you can indeed say:

In atoms and molecules is always a kind of zero-point energy left
  which implies that there is still some 'movement' in the particles.

Now when you ask

So when an electron or quark is really at rest does it still exist?

there is again the idea that being at rest is an absolute notion. But it is not, it depends on the observer. 
So you can see it the other way round: as far as one can say that an electron exists (which is a tricky point if by "existing" one means existing at a definite location at all times), there exists a frame of reference (that is, a way of moving around) in which the electron does not move.

And hence is there really a difference in mass-property of particles or is the equation $E=mc^2$ already suggesting that there is no real property difference in mass
  between a photon and electron? Or there is?

At this point you may now see from the above considerations why I cannot make sense of this question. 
