Is there a name for an element $z$ such that $\,zx = z = xz\,$ for all $x$?

I've heard absorbing element or annihilating element used for this (Wikipedia link).

So in your second example, we'd say that $0$ is an absorbing element for the operation $\times$ on $\mathbb{R}$.

In your first example, you'd have to specify a set in which you're working. I'd suggest formulating this as: for any set $S$, the element $\varnothing\in\mathcal{P}(S)$ is an absorbing element for the operation $\cap\,$.

(2) is a consequence of the ring axioms (see https://en.wikipedia.org/wiki/Ring_(mathematics)); in general, an element $x$ with the property that $x*y=x$ for every $y$ is called an annihilator for the operation $*$.

For (1), the set of subsets of $X$ (for $X$ fixed) forms a Boolean algebra (see https://en.wikipedia.org/wiki/Boolean_algebra; this is the structure provided by intersection, union, and complement). Boolean algebras always have a least element $l$, with the property that $l\wedge x=l$ for every $x$ (where "$\wedge$" is intersection, in the case mentioned above).