Cardinality of sets and strings -> confused
You are confusing the size of a set of things and the size of the things the set contains.
The empty set contains nothing. Its size (or cardinality of you want to be sesquipedalian) is 0.
A set of containing one thing has size 1. It doesn't matter whether the one thing is an elephant, a flea, or a string. If there is one thing, the cardinality is 1. That thing could even be a set, even the empty set. The empty set is a thing which contains nothing but it is not nothing. It is a thing, so it can be the element of another set.
A string is not a set, even though it consists of elements of a set of letters. It differs from a set in that its elements are ordered, and therefore might repeat. When we count the size, we count each repeated element too. But both a string and a set can be empty. The empty set has size 0 but it's also one thing. The empty string has size 0, and it is also a thing. (A different thing).
So,
$$\begin{align}\mid\{\}\mid&amp;=0\\
\mid\{\{\}\}\mid&amp;=1\\
\mid\{“”\}\mid&amp;=1\\
\mid\{“”, \{\}\}\mid&amp;=2\\
\end{align}$$
We usually talk about the length of a string (or other sequence) rather than its cardinality, but the two concepts are shown with the same operator (the pair of vertical bars above, also used for &quot;absolute value&quot;), because they are easily related. In fact, we can think of a string (or sequence) as a set of mappings, each integer in a contiguous finite range is mapped to a single character (or element of the sequence). Even though the two characters are the same, the mappings in $\{&lt;1\to a&gt;, &lt;2\to a&gt;\}$ differ (because the integers differ), and the two mappings clearly denote the string $aa$. The cardinality of the set of mappings is 2, just as is the length of the string.
The empty string corresponds to an empty set of mappings (as one might expect), so its cardinality is 0. (As we saw above, this is different from the cardinality of a set containing the empty string.)

Finally, your question about the pumping lemma is a different confusion, but still a confusion having to do with emptiness.
The pumping lemma states that $|y|&gt;0$, so it is clear that $y$ cannot be  $\epsilon$. But it allows you to pump $y^i$ for any non-negative $i$. There is no contradiction here. $y^0 = \epsilon$ for any $y$, by definition. So the pumping lemma lets you produce a string where $y$ has been replaced with $\epsilon$ (in other words, deleted). That has nothing to do with the fact that the $y$ being eliminated actually existed in the original string. (Or perhaps it does, since eliminating nothing is a no-op.)