To what extent does the distributive axiom sculpt the face of ring theory? How much of ring theory arises simply from applying group-theoretic results to the additive structure, or semigroup/monoid-theoretic results to the multiplicative structure? To be more precise, can anyone describe how much rigidity is imposed on the structure of rings by the distributive axiom which "connects" addition and multiplication? What are some examples of results, say, where distributivity really plays a key role?
 A: One example of where the distributivity axiom adds additional properties to ring-like structures is the following: 
The ring axioms with the multiplicative identity axiom and without the commutativity of addition axiom can already prove the commutativity of addition. 
This requires the distributive property. 
(1 + 1)(x + y) = (1 + 1)(x) + (1 + 1)(y) = x + x + y + y
(1 + 1)(x + y) = 1(x + y) + 1(x + y) = x + y + x + y
so $x + y = y + x$. 
So distributivity forces the commutativity of addition in the case that the ring has a multiplicative identity. 
A: 
How much of ring theory arises simply from applying group-theoretic results to the additive structure, or semigroup/monoid-theoretic results to the multiplicative structure?

Almost nothing. And in any case this would not be called ring theory. For example, a finite subgroup of the multiplicative group of a field is cyclic. This belongs to group theory, not to field or ring theory, although it is crucial for field theory.

To be more precise, can anyone describe how much rigidity is imposed on the structure of rings by the distributive axiom which "connects" addition and multiplication? What are some examples of results, say, where distributivity really plays a key role?

There is no field $K$ with $K^* \cong \mathbb{Z}$.
This kind of rigidity arises everywhere. Just take any interesting result on rings and you will see that it makes no sense at all to try to prove this for pairs of (abelian group,monoid). So, just open a textbook and you will see lots of evidence that distributivity plays a key role.
