Generalization of a ring?

Absolutely! Such things are studied in a few related disciplines encompassing universal algebra and parts of category theory (including what is called "higher-dimensional algebra"). On the categorical side some keywords here include (algebras over an) operad, (algebras over a) monad, and (models of a) Lawvere theory. More concretely there are also coalgebras, bialgebras, Hopf algebras, Frobenius algebras, etc. Ross Street's Quantum Groups: a path to current algebra was written as an introduction to these more exotic types of algebra that people don't generally bother to tell you about as an undergrad. Far from being esoteric generalizations of ordinary algebra, these structures turn up all over mathematics in the most unexpected places.

As a matter of introduction, however, I don't think that what you've just described is a good way to get a sense of what a ring actually is. The ring axioms capture the abstract properties that symmetries of abelian groups satisfy, in the same way that the group axioms capture the abstract properties that symmetries of sets satisfy. The point here being that if $A$ is an abelian group and $f : A \to A$ some endomorphism of it, then there are two ways that one can combine such endomorphisms: one can compose them or add them. This leads to the multiplication and addition operation, respectively, on the ring $\text{End}(A)$ of endomorphisms of $A$. For example if $A$ is the abelian group $\mathbb{Z}$, then $\text{End}(A)$ is the commutative ring $\mathbb{Z}$.

Then there is a "Cayley's theorem" for rings which says that any abstract ring $R$ can be realized as endomorphisms of some abelian group: in fact $R$ is precisely the ring of endomorphisms of $R$ which respect right multiplication, that is, which respect the right $R$-module structure on $R$.

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.