The first claim is established in
R3923: Standard gaussian random real number is symmetric about zero. For the second claim, we have
\begin{equation}
\mathbb{P}(X X \geq 0)
= \mathbb{P}(X^2 \geq 0)
= 1
\end{equation}
and
\begin{equation}
\mathbb{P}(Y X \geq 0)
= \mathbb{P}(- X^2 \geq 0)
= 0
\end{equation}
$\square$
