If $Y \in \mathbb{R}^n$ is random, and $X \in \mathbb{R}^{n \times p}$ is fixed, what is $Var(Y - X(X^TX)^{-1}X^TY)$? If $Y \in \mathbb{R}^n$ is random, and $X \in \mathbb{R}^{n \times p}$ is fixed, what is $Var(Y - X(X^TX)^{-1}X^TY)$?
\begin{align*}
Var(Y - X(X^TX)^{-1}X^TY) &= Var(Y) + X(X^TX)^{-1}X^TVar(Y)X(X^TX)^{-1}X^T - 2Cov(Y, X(X^TX)^{-1}X^TY)\\
&= Var(Y) + X(X^TX)^{-1}X^TVar(Y)X(X^TX)^{-1}X^T - 2X(X^TX)^{-1}X^TCov(Y, Y)X(X^TX)^{-1}X^T\\
&= Var(Y) - X(X^TX)^{-1}X^TVar(Y)X(X^TX)^{-1}X^T\\
\end{align*}
Is the above correct?

No you cannot simplify the variance as you have done.  Perhaps the simplest method is to note that one can we-rewrite
\begin{eqnarray*}
Y - X\left(X^{\prime}X\right)^{-1}X^{\prime}Y = \left[I_n - X\left(X^{\prime}X\right)^{-1}X^{\prime}\right]Y.
\end{eqnarray*}
Therefore, we have that
\begin{eqnarray*}
\mbox{Var} \left(Y - X\left(X^{\prime}X\right)^{-1}X^{\prime}Y\right) &=& \mbox{Var} \left(\left[I_n - X\left(X^{\prime}X\right)^{-1}X^{\prime}\right]Y\right) \\ &=&
\left[I_n - X\left(X^{\prime}X\right)^{-1}X^{\prime}\right] \mbox{Var} \left(Y\right) \left[I_n - X\left(X^{\prime}X\right)^{-1}X^{\prime}\right].
\end{eqnarray*}
