Why Right-Division of regular language with RE\E language is regualr?
In fact, if $L_1$ is regular then $L_1/L_2$ is regular for any $L_2$.

Indeed, consider a DFA for $L_1$ with transition function $\delta$ and accepting states $F$. We modify the set of accepting states to
$$
F' = \{ q : \delta(q,w) \in F \text{ for some } w \in L_2 \}.
$$

You might object that "$F'$ cannot be computed". So what? We never said that you can compute a DFA for $L_1/L_2$ given one for $L_1$ and some reasonable representation of $L_2$. All we claim is that $L_1/L_2$ is regular.

Check out this question for a similar but simpler situation.