What is the cardinality of a lone element? I know that elements which are not sets do not have cardinality, but hear me out.
Suppose that $A=\{u,\{\{v,w\},x,\{y,\{z\}\}\}\}$. The nesting of the sets allows you to uniquely identify particular elements by there "depth" in $A$. For example $z$ is the only element $e\in^4A$ ($\in^4$ is an abuse of notation meant to suggest that $e$ is an element of an element of an element of an element of $A$).
However, this type of statement is not sufficient for uniquely identifying any element other than $z$, as, for example, the set of all $e\in^2 A$ contains $\{v,w\}$, $x$, and$\{y,\{z\}\}$.
It is still possible to distinguish between each of these elements, though; $e\in^2A\ and\ |e|\neq2$ defines $x$, $e\in^2A\ and\ \nexists\ b\in e:|b|=1$ defines $\{u,w\}$, and $e\in^2A\ and\ \exists\ b\in e:|b|=1$ defines $\{y,\{z\}\}$.
The statement used to identify $x$ is true because $x\in^2A$ and $|x|\neq2$, the latter because $x$ is not a set, and therefore cannot have a cardinality of 2. Similarly, $\{y,\{z\}\}\in^2A$, and $|\{z\}|=1$.
Now, since I've used cardinality to differentiate the elements $e\in^2A$ from one another, I should like to define the cardinality of each such element. While this is no problem for the sets $\{u,v\}$, $\{y,\{z\}\}$, and the singleton set $\{z\}\in^3A$, it is an issue for the elements $x$ and $y$. The statement $|x|\neq1$, while true, cannot be evaluated if $x$ does not have cardinality. Since $\{x\}\neq x$, the cardinality of $x$ cannot be 1.
One possibility is that the cardinality of $x$ is 0, but this runs the risk of equating the element $x$ with the empty set, which would violate the axioms of ZFC if $x$ is considered to be a set. On the other hand, you could say that the empty set is the only set whose cardinality is zero, and that $x$ is not a set: thus, $|x|=0\not\implies x=\emptyset$. But this runs into the problem of defining what a set is.
So, what is the best way to define the cardinality of an element? If there isn't one, then is there a good way to describe how a lone element is unlike a set, in set theoretic terms?
 A: In ZF(C) and most other set theories, every object is a set, so in particular each of the elements $u,v,w,x,y,z$ in your question are sets, which have cardinalities by virtue of the fact that they are sets.
There are some set theories such as ZFA and NFU that admit atoms (a.k.a. urelements), which are objects that are not sets. In these settings, you would typically not define $|a|$ when $a$ is an atom.
You could define $|a|=0$ if $a$ is an atom, since atoms don't have any elements. In this case, you're right that the sentence $\forall x,\, |x|=0 \Rightarrow x = \varnothing$ is no longer true. When dealing with atoms, though, it is common to have some way of discerning between sets and atoms, e.g. via a unary predicate $S$ or a definition of a class $V$ of (non-atomic) sets, in which case the sentence $\forall x \in V,\, |x| = 0 \Rightarrow x=\varnothing$ is true. Thus anything you'd want to prove about sets in a regular set theory without atoms, you just relativise to the class $V$ of all sets.
