Why do particles move? Is all particles moving and forever?
Can the movement of particles be stoped ?
 A: I don't know exactly what you are asking. Anyways, it is impossible for any particle to stop moving completely. If it stops the uncertainty principle of quantum mechanics will be violated since in that case both position and its conjugate momentum of the particle will be known with certainty. therefore all particles have to have minimum amount of movement in accordance with the uncertainty principle. For an harmonic oscillator this minimum movement contributes to the zero point energy.
A: Movement itself is a relative quantity. A particle at rest in one frame of reference is moving at a constant velocity in another frame of reference. Rest is never absolute.
In many ways, particles will never stop moving. Not only is the universe undergoing expansion,but as long as energy is fundamental to our universe, movement should not cease. In another light, without "motion", the meaning of time would break down, as we would have no way of telling the past from the future. 
Theoretically speaking, particles move so time can exist, or vice versa.
A: Orbital electron in hydrogen atom is moving forever. Orbital electrons are moving in atoms even at a very low temperature in Bose-Einstein condensate.
A: As long as fields exist, gravitational at any range of distance, and electromagnetic field at short range, the particles will keep moving. 
But the particles themselves are the sources of the fields, and IMO, under modern theoretical grounds the particles 'are' fields. 
So, as particles exists => fields exists => and interaction at a distance exists => they move. 
Interaction is a sort of communication between particles: 'come here' or 'get lost' are the messages. The other way of communication is thru exchange of energy packets: 'photons'. 
The accepted theoretical particle for gravitational exchange is the graviton. But GR does not imply it, as space itself is enough to convey the message 'come here'.
We can only know the momentum or position of any particle thru a measurement, and this is only possible thru an interaction. Actually all measurements that I know of are thru photons exchange that carry away energy from one system/particle to another.
So, we see that particles are condemned to say, each one to all others, 'come here' or 'get lost'.
Now I will try to see what can be a immobile world: lets put the particles far way one from the others, to consider just the gravitational interaction, and so 'frozen' that we can say : the temperature is 0, equivalent to say that the kinetic energy is zero. Absolutely homogeneous is this world that each particle sense the same attracting force from each direction and would stay imobile forever in the same 'location'. 
Is this immobile world, with no definable time, stable forever? No, because the slightest deviation of one particle from his position, due to any quantum fluctuation, will lead, with time, to a circus on all others.
The Time will start now for the particles in the immediate vicinity of that bastard revolutionary particle, the one that moved away from his correct position, because they will start moving away from the hole left in the vacant position. (what happens to a stocking made of nylon when gets with a loose mesh? the hole widens)
Finally, the only possible conclusion is: particles are condemned to motion,  as long as they  exist.
