Yes, any algebraic structure that has multiple operations will have laws like the distributive law that intertwine the operations. For otherwise the operations would not interact in any way and the structure could be studied as two independent structures with the non-interacting operations. For example, if we dropped the distributive law from the ring axioms then we'd simply have a set with a given abelian group structure and given monoid structure with no connection between the two structures. It is the distributive law that ties together these two structures and leads to the rich structure that is unique to rings - structure above and beyond the constituent structure of the additive group and multiplicative monoid.
One can observe the key role played by the distributive law even in the simplest results on rings. For example, consider the proof of the law of signs $\rm\ (-A)\:(-B) = A\:B\:.\ $ One simple proof is to observe that both terms are additive inverses of $\rm\ (-A)\:B\ $ hence they are equal by uniqueness of inverses. But to verify that they are inverse requires applying the distributive law. Similarly, any theorem that is truly ring-theoretic result $\:$ (i.e. $\:$ is not merely a result about abelian groups or monoids) must employ the distributive law in its proof (though perhaps obscured in some remote lemma).
Analogous remarks hold true for any algebraic structure with multiple operations, e.g. lattices with their intertwining absorption law $\rm\ X = X \vee (X \wedge Y)\ $ and it's dual, or distributive lattices (e.g. Boolean algebras) with their distributive law $\rm\ X \vee (Y \wedge Z) = (X\vee Y)\wedge (X\vee Z)\ $ and its dual or, more generally, the important modular law $\rm\ (X\wedge Y)\vee(Y\wedge Z) = Y \wedge ((X\wedge Y)\vee Z)\:.\ $
The common properties of algebraic structures are studied in universal algebra (or general algebra). For example, one major theme is the study of the role played by properties of the lattices of congruences, e.g. congruence lattices of lattices are distributive, and congruence lattices of groups and rings are modular. These properties play fundamental roles in the theories of these structures.