If there is a precipitous ideal $I$ on $\omega_1$---a hypothesis equiconsistent over ZFC with the existence of a measurable cardinal---then after forcing with $P(\omega_1)/I$, we get an elementary embedding $j:V\to M\subset V[G]$ with critical point $\omega_1$. Thus, on this hypothesis, $\omega_1$ is outer measurable in the sense you have described. This answers question 1 and also shows that your assertion that
ZFC proves that a cardinal is outer-measurable only if it is measurable.
is not correct if these large cardinal hypotheses are consistent. Similar phenomenon arise with analogues of many of the other large cardinal concepts, such as supercompactness and others. Matt Foreman has particularly emphasized the richness of these generic large cardinal concepts, and he has explored this topic extensively. Some of this material is in his chapter in the Handbook of Set Theory.
Perhaps you were thinking of the related assertion, which I believe is correct (but this should be confirmed by inner-model theorists), namely, if $\kappa$ is outer measurable, then there is an inner model where $\kappa$ is measurable. That is, if $\kappa$ is the critical point of a generic embedding, then I believe one can still construct the canonical inner model $L[\mu]$ in which $\kappa$ is measurable.
Finally, you seem to indicate that you believe that the concept of outer-measurability is not first-order expressible in ZFC. But I think it is expressible: $\kappa$ is outer measurable if and only if there is a partial order $\mathbb{P}$ forcing that there is a $V$-ultrafilter $\mu$ on $\kappa$ such that $\text{Ult}(V,\mu)$ is well-founded.