What is the $\operatorname{Var}(XY|Y)$ given that $X$ and $Y$ are independent? What is the $\operatorname{Var}(XY|Y)$ given $X$ and $Y$ are independent?
Is it :
$$\operatorname{Var}(XY|Y)= Y^2\operatorname{Var}(X|Y) = Y^2\operatorname{Var}(X)?$$

Your reasoning flow shows that you had a good understanding on conditioning.  Formally, you can derive it from the definition$^\dagger$ of conditional variance and using basic properties of conditional expectations:
\begin{align}
 & \operatorname{Var}(XY|Y) \\
=& E[X^2Y^2 | Y] - (E[XY|Y])^2 \tag{definition}\\
=& Y^2E[X^2|Y] - (YE[X|Y])^2 \tag{pulling out known factors} \\
=& Y^2E[X^2] - Y^2(E[X])^2 \tag{independence}\\
=& Y^2\operatorname{Var}(X).  
\end{align}

$^\dagger$ For two random variables $\xi$ and $\eta$ such that $E[\xi^2] < \infty$, the conditional variance of $\xi$ given $\eta$ is
\begin{align}
\operatorname{Var}(\xi|\eta) = E[(\xi - E[\xi|\eta])^2|\eta] = E[\xi^2|\eta] -
(E[\xi|\eta])^2.
\end{align}
