Should a ring be closed under multiplication?

A ring is an abelian group $R$ with an additional operation $\times$, that is, a function $\times:R\times R\to R$, satisfying the various axioms. The fact that this function has codomain $R$ is exactly the fact that $R$ is closed under multiplication.

I suspect you are misreading the definition. The definition I know says that multiplication is a "binary operation" on the underlying set which means that it "$ab$" is a mapping from $X \times X\rightarrow X$. That is, the result of "$ab$", for $a$ and $b$ two members of the ring is again a member of the ring. I.e. it is closed under multiplication.

Yes, by the definition of a ring.

A ring $(R,+,\cdot)$ satisfies eight properties: five of which is that $(R,+)$ is an abelian group (closed, associative, existence of $0$, invertibility of each element, and commutativity), two of which is that $(R,\cdot)$ is a semi-group (closed and associative) and the eighth property is the distributive property.