Doubts on the mathematical content of the Hitler Downfall Meme.

The null set is just $\emptyset$, the empty set. This is open by definition (and closed too as its complement ($X$, the whole space) is also open (by definition). Normally a null set is reserved for measure theory, and a zero-set is a notion from later in topology: a set of the form $f^{-1}[\{0\}]$ for some real-valued continuous function on $X$.

The null set is not the space they're talking about, AFAIKT.

A clopen set is indeed a set that is both open and closed, it's a usual term (we even use it in Dutch normally, though an older professor of mine preferred "opgesloten" as a Dutch alternative ("gesloten"=closed, "open" = open), but that already means "locked up", so sounds confusing to me personally).

I think "close point" is just a mistranslation of "adherent point" ($x$ is an adherent point of $A$ whenever all neighbourhoods of $x$ intersect $A$) and a set is closed if it equals the set of all its adherent points. "adherent" etymologically means something like "glued to", "clinging to" (from Latin adhaerere.)

The empty set (in that last definitional view) is closed because it has no adherent points, so it indeed contains all its adherent points as a vacuous truth.

I think it's cleaner to define closed sets directly after the introduction of open sets as their complements. But then having notions of interior and adherence and boundary points can be handy as well, and all are easily defined in terms of open sets.

1. It's a synonym for the empty set, $\emptyset$. (Not recommended, since it means something else in measure theory.)
2. No (see previous answer).
3. It says “all the zero of the points in the null set have a neighborhood...”, which I think is supposed to mean “all the points in the empty set (and there are no such points, their number is zero, there is zero of them) have a neighborhood...”. And this is a vacously true statement, which can be confusing for beginners; if there are no points in the set $S$, then any statement of the form “for all $x \in S$ this or that holds” counts as true, since there are no points in $S$ that could falsify it. The clopen set that they are talking about is the empty set.
4. No idea. Maybe a direct translation of a topological term from some other language? (Boundary point, perhaps?)