Maybe it is implicit in your use of parentheses, but I'd like to just point out that the scheme property is inherited by the infinitesimal fibers without any hypotheses on the algebraic space. More generally, if $S$ is an algebraic space such that $S_{\rm{red}}$ is a scheme then $S$ is a scheme. (This is not so easy to prove when $S$ is not locally noetherian, but it is true.)

The precise meaning with which this is actually useful to prove things rigorously over $\mathbf{Z}(\!(q)\!)$ is via Raynaud's construction over $\mathbf{Z}[\![q]\!]$ (deformation from $q=0$!) via algebraization of formal schemes. It provides a canonical identification of the formal completion along the identity section with $\widehat{\mathbf{G}}_m$ (intrinsically characterized too) such that the formal $\mu_N$'s globalize. The invariant 1-form ${\rm{d}}t/t$ on that formal group comes from a global 1-form on the algebraized Tate curve, namely ${\rm{d}}x/(2y+x)$. That is the link.

Dear prochet: You've been asking a lot of "basic" algebraic geometry questions recently. Do you have anybody you can speak with about them in person? Anyway, the answer is negative for a silly reason because you omitted the hypothesis that $f$ is surjective (consider $X$ empty, or $Y$ disconnected with $f$ only hitting one connected component, etc.) If $f$ is surjective then the answer is affirmative in much greater generality. See EGA IV$_4$, 17.7.5.

Yeah, 10.1--10.4 in IV$_3$.

Should have written IV$_3$ above.

There is something deeply ironic about asking for the Jacobson property of the topological space which you already denote by what is actually its set of closed point. Anyway, just read 10.1 in EGA IV$_4$ for an affirmative answer (use the criterion in 10.1.2(a)).

Sure, just use Chevalley's theorem that the image of any finite type map between noetherian schemes is constructible (in the space of the entire target scheme, not just on its subspace of closed points!). Since a constructible subset of $S$ containing all closed points is the entire space (proof: reformulate via emptiness of its constructible complement!), that's it. More generally, for any Jacobson scheme $S$ and its subspace of closed points $S^0$, $A \mapsto A\cap S^0$ is an inclusion-preserving bijection between the sets of constructible subsets of $S$ and $S^0$.