Are $\emptyset$ and $X$ closed, open or clopen?

They are all both open and closed.

Let me make this a bit more clear. By definition of a topology (from wikipedia):

A topological space is then a set $X$ together with a collection of subsets of $X$, called open sets and satisfying the following axioms:

• The empty set and $X$ itself are open.
• Any union of open sets is open.
• The intersection of any finite number of open sets is open.

So just from the definition itself it follows that $∅$ and $X$ are open.

Furthermore a set is closed (by definition) if the complement is open. Therefore $∅$ and $X$ are closed (they are each others complement).

The term clopen means that a set is both open and closed, so they are both also clopen.

By the first axiom in the definition of a topology, $X$ and $\emptyset$ are open. However, closed sets are precisely those whose complements are open, by definition. Hence the empty set and $X$, being each others complements, are also closed. So, they are clopen.

To put it simply: sets are not doors! Being open does not mean that the set is not closed, and being closed do not implies that the set is not open. Yes, the empty set and the whole space are both open and closed. Another more dramatic example is: take a metric space $X$, with the discrete metric. Then every singleton $\{a\}$ (in fact, every subset of $X$) is clopen. Balls $B(a,r)$ with radius $r < 1$ are contained in $\{a\}$, hence $\{a\}$ is open. And $\{a\}$ is also closed, because it's complement is $X \setminus\{a\} = \bigcup_{b \in X, b \neq a}\{b\}$, a union of open sets, hence open.