How do you translate these 2 regular languages in words correctly?
For 1. it is not quite the reverse; it happened to be a reverse in your example. So if $u = aab$ and $v = abb$ and $uv \in L$, then you know $vu = abbaab \in \text{cycle}(L)$, but you wouldn't be able to say that $bbabaa = v^Ru^R \in \text{cycle}(L)$.

For 2. it is not that you want the concatenation of two strings of equal length, but rather you only want the first half of even-length strings in $L$. That is, if, say, $aabbabbabb \in L$, then $aabba \in L/2$.