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Definition 1. An affine space morphism is a smooth, faithfully flat morphism whose geometric fibers are affine spaces.

Definition 1. An affine space morphism is a separated, smooth morphism whose geometric fibers are affine spaces.

Kollár's Flatness Criteria for normal target. Question 12 has a positive answer by Kollár's Flatness Criteria.

MR1339664 (96j:14010)
Kollár, János
Flatness criteria.
J. Algebra 175 (1995), no. 2, 715–727.
https://ac.els-cdn.com/S0021869385712094/1-s2.0-S0021869385712094-main.pdf?_tid=c9850a97-a4a0-4482-84e6-2e5b863bce01&acdnat=1551275145_19dbfad83504d3662477ba2257220fb8

Proposition 13. Every separated, finitely presented morphism whose geometric fibers are affine spaces of constant dimension $n$, whose domain is a finitely presented, integral $k$-scheme, and whose target is a finitely presented, integral, normal $k$-scheme is an affine space morphism.

Proof. Such a morphism satisfies the hypotheses of Corollary 11 of Kollár's article. QED

As mentioned in the comments below, using de Jong's Alterations of Singularities Theorem, for every separated, reduced, finitely presented $k$-scheme $Y$, there exists a proper hypercovering $Y_\bullet\to Y$ of étale cohomological descent such that every term of $Y_\bullet$ is $k$-smooth. This is Theorem 3.1 of the following.

Since $f$ is smooth, by the Smooth Base Change Theorem, also the pullback $X_\bullet := X\times_Y Y_\bullet$ is a hypercovering of $X$ of étale cohomological descent. Thus, we can compute the pullback map from the étale cohomology of $Y$ to the étale cohomology of $X$ in terms of the simplicial pullback maps $H^*(Y_\bullet,\mathcal{E}) \to H^*(X_\bullet, f^*\mathcal{E})$. For every $Y_i$ in the hypercovering $Y_\bullet$, the morphism $X_i\to Y_i$ is in the given class with smooth target. Thus, by hypothesis, the pullback map is an isomorphism. QED

Proposition 11. The pullback map $H^1(f^*,\mu_n)$ is surjective with rank $1$-kernel $\mu_n(k)$, and the pullback map $H^2(f^*,\mu_n)$ is surjective with rank $1$-kernel $\mathbb{Z}/n\mathbb{Z}.

As mentioned in the comments below, using de Jong's Alterations of Singularities Theorem, for every separated, reduced, finitely presented $k$-scheme $Y$, there exists an proper hypercovering $Y_\bullet\to Y$ such that every term of $Y_\bullet$ is $k$-smooth. This is Theorem 3.1 of the following.

By Deligne, a proper hypercovering is of étale cohomological descent for constructible $\mathbb{Z}/n\mathbb{Z}$-modules with $n$ prime to the characteristic of $k$.

Since $f$ is smooth, by the Smooth Base Change Theorem, also the pullback $X_\bullet := X\times_Y Y_\bullet$ is a proper hypercovering of $X$, and thus of étale cohomological descent. Thus, we can compute the pullback map from the étale cohomology of $Y$ to the étale cohomology of $X$ in terms of the simplicial pullback maps $H^*(Y_\bullet,\mathcal{E}) \to H^*(X_\bullet, f^*\mathcal{E})$. For every $Y_i$ in the hypercovering $Y_\bullet$, the morphism $X_i\to Y_i$ is in the given class with smooth target. Thus, by hypothesis, the pullback map is an isomorphism. QED

Proposition 11. The pullback map $H^1(f^*,\mu_n)$ is surjective with rank $1$-kernel $\mu_n(k)$, and the pullback map $H^2(f^*,\mu_n)$ is surjective with rank $1$-kernel $\mathbb{Z}/n\mathbb{Z}$.

Pullback maps are isomorphisms for affine space morphisms to separated, finite type $k$-schemes. Every affine space morphism with reduced target is a relative affine space after base change by an étale morphism with dense image.

Proposition 6. A smooth morphism $f$ to a separated, finitely presented $k$-scheme $Y$ is a universal étale cohomological equivalence if and only if for every point $y$ of $Y$, the fiber over $\text{Spec}\ \kappa(y)^{\text{sep}}$ is an étale cohomological equivalence.

Pullback maps are isomorphisms for affine space morphisms to separated, finite type $k$-schemes. Although the application is to affine space morphisms, it is convenient to formulate the next results in terms of étale cohomological equivalence.

Proposition 6. A smooth morphism to a separated, finitely presented $k$-scheme $Y$ is a universal étale cohomological equivalence if and only if for every point $y$ of $Y$ the fiber over $\text{Spec}\ \kappa(y)^{\text{sep}}$ is an étale cohomological equivalence.