Reduction of elliptic curves defines over general field

Question

let $(K,v)$ be a field with a non-archimedean valuation and let $E/K$ an elliptic curve. The field $K$ is not supposed to be local. We denote by $\mathcal{O}_v$ the valuation ring (not a DVR) and $M_v$ the maximal ideal.

1) I'm looking for a definition of the reduction of the elliptic curve $E$ at $v$, it is just the reduction of the coefficients of the Weierstrass equation by the map $\mathcal{O}_v \mapsto \mathcal{O}_v / M_v$ ?

2) In my case, the field $K$ is not local, so $\mathcal{O}_v$ is not a discrete valuation ring but can I have a Néron model $\mathcal{E}$ of $E$ defined over $\mathcal{O}_v$ ?

Thanks for answers !