Regularity of the exact middle of words from a regular language
Hint: Consider some DFA for $L$. For every $n \geq 0$, let $A_n$ be the set of states $s$ such that there is some word of length $n$ which leads the DFA from the initial state to $s$. Let $B_n$ be the set of states $t$ such that there is some word of length $n$ which leads the DFA from $t$ to an accepting state. Finally, for any two states $s,t$, let $R_{s,t}$ be the (regular) set of words leading the DFA from $s$ to $t$. We have
$$ L_2 = \bigcup_{n \geq 0} \bigcup_{\substack{s \in A_n \\ t \in B_n}} R_{s,t}. $$
Since there are only finitely many possibilities for $s,t$, the union is in fact finite, and so regular.