Question

In a head-on collision between two particles, is it necessary that the particles will acquire a common velocity at least for one instant?

Answer 1

Yes.
For example, consider particle-1 at a velocity of 4 ms^-1 and particle-2 at a velocity of 2 ms^-1 undergo a head-on collision.
The velocity of particle-1 decreases but particle-2 increases. Therefore, at an instant, their velocities will be equal.