Is projecting into a set $X$ and then projecting the result into the second set $Y$ the same as projecting into the intersection of $X$ and $Y$?
I am not sure if these information will be needed to answer my question, but just to provide some contextual details.
Let's assume $X=\{x : Ax=0\}$ and $Y=\{y: y\geq 0\}$.
No, not in general. Try imagining two non-orthogonal, non-parallel lines in $\Bbb{R}^2$ (which obviously intersect at a single point). For most points, projecting onto one line, then onto the other, will not even produce a point in the intersection, let alone the closest such point!