I would say yes. If the connectivity between $x$ and $y$ has decreased, then obviously the connectivity of $G$ has decreased. On the other hand, if $G - e$ is not 3-connected, then there are a pair of subgraphs $(A,B)$ such that $G=A \cup B$, and $|V(A) \cap V(B)|=2$. Since $G$ is 3-connected, this implies that $e$ must have one endpoint in $V(A)-V(B)$ and the other in $V(B)-V(A)$. Hence the connectivity between $x$ and $y$ has indeed decreased in $G-e$.

I would say yes. If the connectivity between $x$ and $y$ has decreased, then obviously the connectivity of $G$ has decreased. On the other hand, if $G - e$ is not 3-connected, then there are a pair of subgraphs $(A,B)$ such that $G-e=A \cup B$, and $|V(A) \cap V(B)|=2$. Since $G$ is 3-connected, this implies that $e$ must have one endpoint in $V(A)-V(B)$ and the other in $V(B)-V(A)$. Hence the connectivity between $x$ and $y$ has indeed decreased in $G-e$.

I would say yes. If the connectivity between $x$ and $y$ has decreased, then obviously the connectivity of $G$ has decreased. On the other hand, if $G - e$ is 2-connected, then there are a pair of subgraphs $(A,B)$ such that $G=A \cup B$, and $|V(A) \cap V(B)|=2$. Since $G$ is 3-connected, this implies that $e$ must have one endpoint in $V(A)$ and the other in $V(B)$. Hence the connectivity between $x$ and $y$ has decreased in $G-e$.

I would say yes. If the connectivity between $x$ and $y$ has decreased, then obviously the connectivity of $G$ has decreased. On the other hand, if $G - e$ is not 3-connected, then there are a pair of subgraphs $(A,B)$ such that $G=A \cup B$, and $|V(A) \cap V(B)|=2$. Since $G$ is 3-connected, this implies that $e$ must have one endpoint in $V(A)-V(B)$ and the other in $V(B)-V(A)$. Hence the connectivity between $x$ and $y$ has indeed decreased in $G-e$.