Let $A=S^{-1}\mathbb Z$ be a localization of a multiplicative set $S\subset \mathbb Z$.

Question 1: Let $X$ be an arbitrary $A$-scheme, and view $X_A=X\times_{\mathbb Z} A$ as an $A$-scheme via the second projection. Is there a canonical isomorphism of $A$-schemes $X_A\cong X$?

More generally, let $R=a_1\mathbb Z\oplus\dotsc\oplus a_n\mathbb Z$ be a commutative unital ring and let $A=S^{-1}R$ be a localization of a multiplicative set $S\subset R$.

Question2: Let $X$ be an arbitrary $A$-scheme, and view $X_A=X\times_{\mathbb Z} A$ as an $A$-scheme via the second projection.

a) Let $X^n$ be the disjoint union of $n$-copies of $X$. Is there a canonical isomorphism of $A$-schemes $X_A\cong X^n$?

b) Is it possible to compute $Hom_{\mathbb Z}(A,X)$ in terms of $Hom_A(A,X)$ and $n$?

c) Moreover, is $Hom_{\mathbb Z}(A,X)$ in bijection with the disjoint union of $n$ copies of $Hom_A(A,X)$?

I am mainly interested in the case where $R$ is the ring of integers of a finite extension $F$ of $\mathbb Q$, and $A$ is obtained by inverting finitely many prime ideals of $R$; so $A$ is a ring of $S$-integers in the number field $F$. However, I expect the case of general $R,A$ should be similar.

Let $S$ be a finite set of maximal ideals in $ O_K$, where $O_K$ is the ring of integers of some number field $K$. Define $A= O_K[S^{-1}]$.

Let $X$ be an arbitrary $A$-scheme. Consider the scheme $X_A=X\times_{\mathbb Z} A$ as an $A$-scheme via the second projection.

Is $X_A$ the disjoint union of at most $[K:\mathbb Q]$ copies of $X$?

Edit: I rewrote the question to make it more clear.