$X/|X|$ is almost surely a uniformly random element on the unit sphere of dimension $n-1$; this is not the same thing as a rotation, which is a matrix. Since a multivariate standard Gaussian vector is invariant in law under a fixed rotation, its law is certainly invariant under a random rotation as well: thus if $A$ is a uniformly random orthogonal matrix, then $A Y$ is again standard Gaussian. Now let $Z = (X/|X|)^T Y$. Then $Z$ has the same law as the first component of $AY$, which is clearly univariate standard Guassian. To verify its independence with $X$: if you rotate $X$ by an orthogonal matrix $A$, you can absorb $A$ into $Y$, whose law is invariant under rotation. So conditional law of $Z$ under two different $X$ values differing by a rotation stays the same. More obviously, if you scale $X$, the conditional law of $Z$ remains the same.
Another thing I noticed is that $Z$ and $Y$ are not jointly normal! Notice that conditioning on $Y$ clearly has an effect on $Z$. Now assuming they are jointly normal, we can show $E Z v^T Y = 0$ for all vector $v$. In fact we can show $E (X/|X|)^T u v^T u = 0$ for fixed $u,v$. In fact, this follows simply from $E (X/|X|)^T u = E (-X/|-X|)^T u = 0$. This is a contradiction. This is similar to the situation $B X$ and $X$, where $X$ is standard 1d gaussian and $B$ is an independent Bernoulli $\pm 1$ variable. They are not jointly normal either.