Is the empty set a subset of itself?

There is only one empty set. It is a subset of every set, including itself. Each set only includes it once as a subset, not an infinite number of times.

Let $A$ and $B$ be sets. If every element $a\in A$ is also an element of $B$, then $A\subseteq B$.

Flip that around and you get

If $A\not\subseteq B$, then there exists some element $x\in A$ such that $x\notin B$.

If $A$ is the empty set, there are no $x$s in $A$, so in particular there are no $x$s in $A$ that are not in $B$. Thus $A\not\subseteq B$ can't be true. Furthermore, note that we haven't used any property of $B$ in the previous line, so this applies to every set $B$, including $B=\emptyset$.

(From a wider standpoint, you can think of the empty set as the set for which $x\in \emptyset\implies P$ is true for every statement $P$. For example, every $x$ in the empty set is orange; also, every $x$ in the emptyset is not orange. There is no contradiction in either of these statements because there are no $x$'s which could provide counterexamples.)