In the definition of a ring, it is nowhere stated that it must be closed under multiplication. But it seems to be true for all the examples of rings that I've seen so far. So, is this implicitly assumed in the definition or can it be proved?
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.