Topology, closure definition - well defined?

If you're confused about the fundamentals, I'd suggest going over the first chapter in Munkres's book. I'd also recommend the section "Background in set theory" from these notes by Prof. Curtis McMullen, which illuminates the axiomatic foundations of set theory known as the ZFC axioms, the most widely used today.

As you said, the intersection is over a nonempty collection, and therefore is non-trivial. Moreover, it is unique, because there's nothing it could possibly depend on; the subset of closed sets that contain $A$ is a well-defined set of subsets of $X$, so the intersection of all of them is well-defined too (this follows by axiom IV from McMullen's notes, Unions, and taking complements).

If $A\in S$, then $\bigcap S \subseteq A$. Thus by the axiom of separation, $\bigcap S$ exists if $S\ne\emptyset$. Because it can be given as an explicit class builder:

$$\bigcap S=\{x\mid\forall y\in S,x\in y\},$$

it is unique. Thus the intersection of any nonempty collection of sets is a well-defined set. (The intersection of the empty set is also well-defined, but equals the universe, $\bigcap\emptyset=V$, which is not a set.) In topology, though, we want more than that: we want to know that the closure is closed, which follows from one of the axioms of a topology - any arbitrary intersection of a nonempty collection of closed sets is closed.

Sure, let $S = \{ C | C^c \in \tau, A \subset C \}$.

Then $x \in \overline{A}$ iff $x \in C$ for all $C \in S$.