This gives good numerical error control, at the price of positions and velocities being evaluated at different times. One often thinks of thermal particles. The temperature plus the total kinetic energy gives a total energy that is tracked. A random component added to the velocities can, for example, keep the system isothermal. The second question depends on the kind of force you are working with.
Lets begin by considering the rubber particles - particles interact only when they collide.
This short-range force suggests that we divide the entire domain into cells, sort particles into cells, and search for possible collisions only within a particle's own cell and its immediate neighbors. At the boundary, one often uses periodic conditions. Again, this is not a good approach if examining long-range forces.