Proving that $L=\{ w \mid \lvert w \rvert$ is prime $\}$* is a regular language
In fact this follows from a general property of the star operation. When we start with an arbitrary language $L$ of all strings of certain lengths, then the star $L^*$ of that language is always regular. More precisely:

Let $A\subseteq \mathbb N$, and $L = \{w\in \Sigma^*\mid |w|\in A\}$.
Then $L^*$ is regular.

This is a consequence of a property of unary languages, i.e., languages over a single symbol. See this question:
If $L$  is a subset of  $\{0\}^∗$ , then how can we show that  $L^∗$  is regular?
So for $\Sigma = \{0\}$ the statement above is true. For larger alphabets perform an alphabetic substitution (say $\Sigma = \{a,b\}$, then in a regular expression for $L^*$ replace every $0$ by $(a+b)$).