First of all, if the collision is elastic, the distribution of momentum in between the components is completely determined by momentum and energy conservation!
This statement is most obvious in the center-of-mass frame where the total momentum is zero and the two objects are moving in opposite directions. The momentum conservation (the total momentum is zero) implies that they're moving in opposite directions after the collision as well and the ratios of the momenta are the same as they were before collisions. Energy conservation than determines that the absolute normalization of these final momenta has to agree with the initial momenta, too.
So in the center-of-mass frame, only the angles change! In particular, a straight collision of two idealized balls of the same mass has a clear outcome: if we describe the situation from the viewpoint of the table, the cue ball stops and transmits the whole energy to the previously static ball that was hit. This ball we hit is moving with the same speed that previously belonged to the cue ball.
This also pretty much settles the collisions of non-rotating objects without friction and deformation. Cars may be close to this idealization or not, depending on the amount of inner damage.
The deviations of the balls' final speeds from the unique, calculable speeds depends on the additional effects of the angular momentum, friction between the two balls, and friction between the balls and the table. And yes, the friction between the balls and the air which is responsible for some fancier effects not discussed here. In particular, if the ball is rolling forward – which is usually the case when you hit it "slowly" or "gradually" – it will tend to preserve its angular momentum and continue rolling forward. The angular speed $\omega$ of the ball will obey $R\omega=v$ where $v$ is the linear speed of the ball and $R$ is its radius. That corresponds to the frictionless rolling motion of the bottom side of the ball on the table.
If you hit it roughly, it's likely that the ball will be moving at some speed but rotate at a lower speed than what is needed for it to roll on the table without friction. I am not saying that the ball is actually rotating backwards. Instead, it is not rotating at all, so it will never try to revert the direction of motion, as you observed.
But if the cue ball isn't sufficiently rotating forward, it will be more likely to bounce back. The reason is simple. As the cue ball A hits the previously static ball B, the ball B starts to move forward and it also starts to roll forward, because of the friction between B and the table. But by momentum conservation, B has to act on A that starts to roll backwards (the point at which the two balls are meeting is moving up which has the "opposite" implications for the two balls). This backward rotation of the hard cue ball A will tend to send A in the backward motion, too.
All these considerations become much more complicated when the collision isn't straight, when it is left-right-asymmetric.