In kick-boxing, when a fighter's leg hits an opponents leg, the outcome, based on Newton's 3rd law, should be the same for each fighter. It's not even important who kicked who, as in the moment of contact the attacker should feel more less the same as the defender.
You are correct.
As you noted, Newton's third law does indeed say that the force on each fighter's body is the same (but in opposite direction) at each instant in time.
This guarantees not only that the forces are equal, but also the impulse delivered to each of the colliding objects.
Denoting the impulse imparted to objects 1 and 2 as $J_1$ and $J_2$ we therefore have
\begin{align}
F_1(t) &= -F_2(t) \\
\text{and} \qquad J_1 \equiv \int dt \, F_1(t) &= -\int dt \, F_2(t) \equiv -J_2
\end{align}
Impulse has the same dimensions as momentum, so really what we're saying is that in a collision both objects experience the same change in momentum over the same period of time.
Some web pages inform me, that because of the 3rd law, a fighter should make powerful but very brief hits - retracting a kicking leg before it receives a reaction. But from what I know, if you don't feel a reaction, there was no action in the first place.
Again, you are absolutely right.
Retracting the arm or leg does not reduce the force or impulse delivered to that arm or leg during the blow.
How is it ever possible to take advantage in martial arts hit and win?
You actually already partially got it when you mentioned hardness of the objects involved in the collision.
The technical words for describing this are stress and strain.
Stress is essentially the inter-molecular or inter-atomic forces within a solid.
Strain is the deformation of the solid from its usual shape.
When an arm or leg hits a nose, both experience the same impulse, but because the nose is softer, it deforms more.
Once the nose tissues move too much relative to one another the the nose breaks.
The elbow, on the other hand, is made largely of calcium and and support much larger internal stress while maintaining low enough strain that the tissues don't move too much relative to one another; e.g. the elbow doesn't break.
Once the collision is over the bone molecules move back to where they were before.
Of course, if the stress in the bone is too large, and consequently the strain exceeds a certain amount, the bone fractures.$^{[a]}$
You can think of the difference between pushing on a nose or an elbow in terms of pushing on springs with different spring constant.
Supposing we have $F = k x$, then for a given force (stress) the displacement (strain) is $x = F/k$.
A low $k$ means a large strain (like the nose) while a large $k$ means less strain (like the elbow).
Of course, there are also biological factors (but which are fundamentally physical of course).
Certain parts of the body are simply more important than others.
An elbow-skull collision does not have equal damaging effects to the owners of said elbow and skull.
The impulse imparted to the elbow causes compression of the bone which transduces the force to articulating structures such as the shoulder.
The skull, on the other hand, transduces the impulse to e.g. the brain.
Rattling a shoulder around may hurt, but rattling a brain around leads to unconsciousness.
$[a]$: There is fascinating data regarding the stress/strain curves for bone. At low stress the bone is essentially elastic. Past a threshold, the strain is a much steeper function of stress. Then at a critical point the bone fractures.
P.S. I've left out a discussion of exactly how/why body tissues are destroyed by excessive strain. In other words, why doesn't your nose just always spring back to its original shape after being deflected by an elbow? A careful description of this process on the microscopic level would make an interesting subject for another question.
