Momentum = (mass) x (velocity)

If the particle is at rest, velocity = 0, and momentum = 0.

Allways...

Acceleration and so a force acting on the particle.

That simply means that its velocity is changing.

yes. If the forces acting on the a moving particle are in equilibrium, (e.g. when a spherical object reaches terminal velocity (neglecting increased air resistance as it gets closer to the ground)) then the particle will be moving at a velocity, that is not 0, yet the velocity will remain constant, and the body will not accelerate or decelerate in any direction, and thus the acceleration is 0.