Are the regular languages closed against injecting single letters?
This language is regular. Let $M$ be a DFA accepting $L$ and assume it has a single initial and final state. We can construct a NFA accepting $L_2$. We create two copies of $M$: $M_1$ and $M_2$. Let us denote the states of $M_1$ as $q_1, \dots, q_n$ and the states of $M_2$ as $p_1, \dots, p_n$. Note that these two DFA are isomorphic where $q_i$ corresponds to $p_i$. The initial state of the new automaton will be $q_1$, and its accepting state will be the accepting state of $M_2$ (so the accept state of $M_1$ is no longer an accept state, and the initial state of $M_2$ is no longer an initial state; in the new NFA, these are just normal states).  Add new transitions $\delta(q_i, a) = p_i$ for each pair $(q_i,p_i)$. The resulting NFA accepts $L_2$.  You can think over it to prove it accepts $L_2$; effectively, $M_1$ is reading the $u$ part of the input and $M_2$ is reading the $v$ part.