The first claim is established in
R3923: Standard gaussian random real number is symmetric about zero. As for the second claim, we have
\begin{equation}
X + Z = X - X
= 0
\end{equation}
surely, as well as
\begin{equation}
Y + Z
= - X - X
= - 2 X
\end{equation}
and clearly $0 \overset{d}{\neq} - 2 X$. $\square$
