How would I properly write down "There exists some $a$ and there exists some $b$ such that $ab = z$."?
Would it be $(1)$ $$\exists a, b: \, ab=z$$ or $(2)$ $$\exists a,\,\exists b:\, ab=z$$ or $(3)$ $$(\exists a)\land (\exists b):\, ab=z$$ or something else?
Both (1) and (2) are fine and widely used. (3) is completely wrong.
(2) is slightly more formal than (1), because in formal logic only (2) (without ":") is a well formed formula. But this is a matter of taste, because the (widely accepted) convention is that (1) is a shortcut for (2). And the use of ":" after an existential quantification $\exists x$ is widely accepted as well, to improve the readability of the expression.
Note that (3) is an abstract nonsense: the connective $\land$ ("and") should be used to connect two propositions, but $\exists a$ and $\exists b$ are not propositions.