Let $E \rightarrow S$ be an elliptic curve (i.e, a smooth proper curve of genus 1).
If $S = \text{Spec (K)}$ where $K$ is a local field, the usual way of doing a reduction at a prime $\mathfrak{p} = (\pi)$ is by reducing the defining equation of the elliptic curve modulo $\pi$ or, equivalently, by finding the Néron model of $E$, namely $\tilde{E} \rightarrow \text{Spec} (\mathcal{O}_K)$ and define $\tilde{E}_{\kappa(\mathfrak{p})} \rightarrow \text{Spec} (\kappa(\mathfrak{p}))$ as the reduction in the prime $\mathfrak{p}$. In the case of $K$ a global field, the same process applies when completing in the prime $\mathfrak{p}$ and reducing to the case of a local field.
And, now comes the questions:
1) It's natural to do another process. Suppose $K$ is a global field. If $\tilde{E} \rightarrow \text{Spec} (\mathcal{O}_K)$ is the Néron model, then I could simply define $\tilde{E}_{\kappa(\mathfrak{p})} \rightarrow \text{Spec} (\kappa(\mathfrak{p}))$ as the reduction, where the pullback is given by the inclusion of a point at $\mathfrak{p}$. Or I could define the reduction by using the Néron model over $\mathcal{O}_{K, \mathfrak{p}}$ and defining the reduction as the fiber over $\mathfrak{p}$. Are these constructions equivalent?
2) It's known that if the reduction is bad, then it's multiplicative or additive. In these cases, respectively, $E_{nonsing} \cong \mathbb{G}_m$ and $E_{nonsing} \cong \mathbb{G}_a$. Is it possible to prove such isomorphisms in a more scheme-theoretical way? In Silverman's book, for instance, the proof uses the defining equation of $E$.
3)Is there a more scheme-theoretical way of defining what's a split multiplicative reduction?
4)If $S$ is an arbitrary scheme and viewing $E \rightarrow S$ as a deformation of elliptic curves, is it possible to define a reduction of this family at some prime under certain suitable conditions? For instance, if the residue field is always the same ($\kappa (s)$ is always the same for each $s \in S$), then it's possible to find the Néron model of each fiber $E_s \rightarrow \text{Spec} (\kappa (s))$ and reduce at a given prime $\mathfrak{p}$ of $\mathcal{O}_{\kappa (s)}$, however I can't see any reasonable way to glue everything (such that the morphism to the base is at least flat).
Thanks in advance.