For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$

The empty set is indeed a set (the set of no elements) and it is a subset of every set, including itself. $$\forall A: \emptyset \subseteq A,\;\text{ including if}\;\; A =\emptyset: \;\emptyset \subseteq \emptyset$$

$$\text{BUT:}\quad\emptyset \notin \emptyset \;\text{ (since the empty set, by definition, has no elements!)}$$

That is, being a subset of a set is NOT the same as being an element of a set: $$\quad\subseteq\;\, \neq \;\,\in: \;\; (\emptyset \subseteq \emptyset), \;\;(\emptyset \notin \emptyset).$$

$\emptyset \;\subseteq \;\{1, 2, 3, 4, 5\},\quad$ whereas $\;\;\emptyset \;\notin \;\{1, 2, 3, 4, 5\},\;$.

$\{3\} \subseteq \{1, 2, 3, 4, 5\},\quad$ whereas $\;\;3 \nsubseteq \{1, 2, 3, 4, 5\}, \text{... but}\; 3 \in \{1, 2, 3, 4, 5\}$.

Being a subset and being an element are different. For every set $A$, $\varnothing\subseteq A$, but not necessarily $\varnothing\in A$. As an example, we can consider $$A=\left\{1,2,3,\{1,2\}\right\}\text{ and }B=\{1,2,3\}.$$ Then $\{1,2\}\in A$ AND $\{1,2\}\subseteq A$ whereas $\{1,2\}\subseteq B$ but $\{1,2\}\notin B$. Hopefully this clarifies that being an element and a being a subset are different things.

$\varnothing$ is a subset of $\varnothing$, but $\varnothing$ is a not an element of $\varnothing$, because $\varnothing$ has no elements by definition.

For every set $A$, there is no element of $\varnothing$ that is not in $A$, and therefore $\varnothing\subseteq A$. This is true whether or not $A$ is empty.

For an example of a nonempty set that doesn't have the empty set as an element, consider the set $A=\{\{1\}\}$. That is, $A$ is the set whose only element is the set $\{1\}$. Because $\{1\}$ contains the element $1$, it is not empty, $\{1\}\neq\varnothing$. Because the only element of $A$ is not $\varnothing$, $\varnothing\not\in A$. The set $\{\{\varnothing\}\}$ also does not have the empty set as an element for the same reason. On the other hand, the set $\{\varnothing\}$ does have the empty set as an element. By definition, $\varnothing$ is the only element of the set $\{\varnothing\}$.