Regular Expressions. Reversal of concatenated strings verification
For the first part,
$$L=\{aw|w\in\{a,b\}^*\}$$
and
$$L_R=\{wa|w\in\{a,b\}^*\}$$

It is true that $L_R$ is the reverse of $L$. $L$ represents the set of strings starting with the symbol $a$, and $L_R$ represents the set of strings ending with the symbol $a$. Note that for some $w$ such that $aw\in L$, you only need to find some $w'$ such that the reverse of $aw$ is $w'a$. You do not need $w=w'$, so your example does not counter the statement.

For every string $w\in\Sigma^*$, where $\Sigma=\{a,b\}$, reversing the characters of $w$ must also belong to$\Sigma^*$. Thus, every string $aw$ in $L$, there is a string which is its reverse in $L_R$ and vice-versa.

For the second part, there is no significance in reversing $v$. It might be intentionally a part of the problem statement.