Some questions regarding decidability and semi-decidability of $A/B = \{ w \text{ | }\exists z \in B, wz \in A\}$
Let $\Sigma$ be your alphabet. You want to design a Turing machine $T$ such that, given any word $w \in \Sigma^*$, $T(w)$ accepts if and only if $w \in A / B$. Notice that since you only want to prove that $A/B$ is semi-decidable, $T(w)$ is not required to reject when $w \not\in A / B$.
Since $A$ and $B$ are decidable, you know that there are two Turing machines $T_A$ and $T_B$ such that that $T_A(x)$ (resp. $T_B(x)$) accepts if $x \in A$ (resp. $x \in B$) and rejects otherwise.
You can then design $T$ as follows:

For each word $z \in \Sigma^*$:

If $T_B(z)$ accepts and $T_A(wz)$ accepts:

Accept.