Robust Variance of a Difference of Independent Variables

Let $X$ and $Y$ be independent random variables with $\operatorname{Var}(X) = 4$ and $\operatorname{Var}(Y) = 9$. A student claims that $\operatorname{SD}(X - Y) = \operatorname{SD}(X) - \operatorname{SD}(Y) = 2 - 3 = -1$. Find the correct value of $\operatorname{Var}(X - Y)$ and $\operatorname{SD}(X - Y)$, and explain the student's error.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.