Closure of quasi projective scheme If $S$ is a scheme, $X$ is a smooth quasi-projective $S$-scheme, is the $S$-projective closure of $X$ a smooth $S$-scheme with $X$ an open subscheme?
 A: I don't know what you mean by the $S$-projective closure of $X$, but for most locally closed immersions¹ $X \hookrightarrow \mathbb P^n_S$, the closure of $X$ in $\mathbb P^n_S$ will not be smooth (even if $S = \operatorname{Spec} k$).
For example, if $k$ is a field of characteristic $p > 0$ and $X$ is a smooth quasi-projective $k$-variety, it is not even known if there exists any smooth proper variety $X'$ that is birational to $X$ (this is a very weak form of resolution of singularities).
It is always true that the map from $X$ to its closure in $\mathbb P^n_S$ is an open immersion. In the Stacks project, this follows from the definition of an ample line bundle on a quasi-compact $S$-scheme, plus part (4) of Tag 01VJ.
Alternatively, you can use that quasi-compact immersions factor as an open followed by a closed [Tag 01QV]. The immersion $X \hookrightarrow \mathbb P^n_S$ is quasi-compact because $X \to S$ is quasi-compact.

¹The existence of an immersion $X \hookrightarrow \mathbb P^n_S$ means that $X \to S$ is what the Stacks project calls H-quasi-projective (H for Hartshorne, as opposed to the more general definition given in EGA). For simplicity, I am sticking to this definition here.
