You really should look at $-Y$ as Normal with mean $-M_y$, variance $\Sigma_y$ and use the formula for $Var (X+Y)$.
Alternatively, for any random variables
$E(X-Y) = EX - EY$ and $Var(X-Y) = Var (X) + Var (Y) - 2 Cov (X,Y)$.
Your expression for Variance is almost always wrong. The only obvious casee I can see it is correct is when $Y$ is a constant or $Y=-X$.
What is interesting here is whether $Z$ is normally distributed. The answer is
Yes if (X,Y) is jointly normal
Not necessarily if (X,Y) are only known to be marginally normal
Consider this construction:
take $X$ to be $N(0,1)$ distribution. Take $Y=X$ if $|X|\leq c$, $Y=-X$, if $|X|>c$, where $c>0$.
Convince yourself $Y$ is also distributed as $N(0,1)$!
here $X-Y=0$ if $|X|\leq c$, else it equals to $2X$, which is never not normal.