Let $\mathbb{A}^2_{\mathbb{Z}}$ the affine plane and it is well known that
$\mathbb{A}^2_{\mathbb{Z}}$ represents the contravariant functor $$\mathbb{A}^2: CRing^{op} \to Set, R \mapsto \{(r,s) \vert r,s \in R\}$$ because $\mathbb{A}^2_{\mathbb{Z}}= Spec \text{ }\mathbb{Z}[X,Y]$ and every map $\mathbb{Z}[X,Y]\to R$ determines a pair $ (r,s) \in R^2$. therefore the antiequivalence of categories $Ring$ and $AffSch$ imples $$Hom_{CRing}(Spec(R),\mathbb{A}^2_{\mathbb{Z}}) = Hom_{CRing}(\mathbb{Z}[X,Y] ,R) \cong \{(r,s) \in R^2\vert r,s \in R\}$$
Questions:
$Q_1$: How to verify that $\mathbb{A}^2$ can be extended to a contravariant functor $Sch/\mathbb{Z} \to Set, S \mapsto \{(r,s) \vert r,s \in \mathcal{O}_X(X)\}$? heuristically I would try to reduce it to known affine case since I can always cover $S$ by affine schemes $U_i :=Spec(R_i)$, i.e. $S= \bigcup_i U_i$.this looks like a situation where the most books on ag finish the proof with the sentence that "everything glues nicely together".
And I would like to know if it is in this situation really so obvious? Namely I choose an arbitrary pair $(r,s) \in \mathcal{O}_X(X)^2$. restricting it to $U_i$ gives compatible family $\{(r \vert _{U_i},s \vert _{U_i})_{U_i}\}$ and appying the sheaf axiom there is a bijection between pairs $(r,s) \in \mathcal{O}_X(X)^2$ and compatible families $\{(r \vert _{U_i},s \vert _{U_i})_{U_i}\}$. these can be threated by the affine case and the conclusion is $\{(r,s) \vert r,s \in \mathcal{O}_X(X)\} \cong \bigcup_iHom_{CRing}(Spec(R_i),\mathbb{A}^2_{\mathbb{Z}})/ \sim$ where "$\sim$" defines the compatibility relation. can I use now some nice property of $Hom$ to conclude that $$ Hom_{CRing}(S,\mathbb{A}^2_{\mathbb{Z}}) \cong \bigcup_i Hom_{CRing}(Spec(R_i),\mathbb{A}^2_{\mathbb{Z}})/ \sim $$?
$Q_2$: If I remove the "point" $(0,0)$ from $\mathbb{A}^2_{\mathbb{Z}}$ then I obtain a new scheme $Z:=\mathbb{A}^2_{\mathbb{Z}} \backslash \{(0,0)\}$ that is not affine.
nevertheless I suppose that $Z$ represents also another interesting functor, namely $$F: CRing^{op} \to Set, R \mapsto \{(r,s) \vert r,s \in R\ \text{ & } r \text{ or } s \in R^*\}$$ now the fundamental problem is that since $Z$ is not more affine I don't know if I can also use an argument as in $Q_1$ via sheaf axiom and glueing. is there another clever way to attack this problem?