What is the cause for the validity of Newton's third law? What, specifically, causes newton's third law?  For instance, if I push on a wall, why is it that I experience a force in the opposite direction?  
I seem to vaguely understand that is has something to do with electronic repulsion or molecular compression (maybe that's completely wrong, I don't know).  As a related question, what would happen if two objects that were /infinitely/ immovable (at the molecular level, it cannot be broken or compressed) collided with each-other?
 A: In John's answer he states: 

But we know that momentum is conserved, ...

So we can derive Newton's third law from the conservation of momentum.
But you may ask where does momentum conservation come from? 
Noether's Theorem
If we apply Noether's Theorem with the condition that the Lagrangian is invariant over space then we get that there is a quantity (momentum) which is conserved.
If you construct your Lagrangian such that it varies from place to place you may get a situation which does not obey the Newton's third law. However this is never necessary and rarely desired.
A: We start by noting that force is the rate of change of momentum. Let's suppose you and I are floating in space (so we are the only two interacting bodies) and you're pushing me so I feel a force $F_{me}$, then:
$$ F_{me} = \frac{dp_{me}}{dt} $$
where $p_{me}$ is my momentum.
But we know that momentum is conserved, so since you are the only thing interacting with me your momentum, $p_{you}$, must be changing in the opposite sense to balance out the changes in my momentum. In other words:
$$ \frac{dp_{you}}{dt} = - \frac{dp_{me}}{dt} $$
And since force is rate of change of momentum that means there is a force on you:
$$ F_{you} = \frac{dp_{you}}{dt} = - \frac{dp_{me}}{dt} = -F_{me} $$
So the two forces are equal and opposite just as Newton's third law tells us.
The details of exactly how the forces are transmitted depend on exactly how the two bodies are interacting, but whatever the interaction the changes in momentum must be equal and opposite, and therefore the forces are equal and opposite.
