stacks - Descend finite etale algebras

I don't think so (finite etale covers cannot be localized in smooth topology in the sense that you describe). Say, $mathcal{X}$ is a point, and $X$ is a smooth variety with non-trivial fundamental group, say, an elliptic curve (or $mathbb{A}^1-{0}$). Then $pi$ is a presentation. Let $f:Yto X$ be a non-trivial finite etale cover, say, the cover of the elliptic curve by an isogeneous elliptic curve. Then your question becomes: `is there a trivial (i.e., lifted from $mathcal{X}$) cover $Y'$ of $X$ with a map to $Y$? This is of course not true.