Regarding (1), since the localisation map from $\mathbb{Z}$ to $\mathbb{Z}_{(p)}$ is injective, and $MU_\ast$ is free over $\mathbb{Z}$, localisation $MU \to MU_{(p)}$ will be injective, and so the image of the localisation map will again be $MU$.
Regarding $(2)$, localisation does preserve complex orientations -- a class $u \in E^2(\mathbb{C} P^\infty)$ is an orientation if its restriction to $S^2$ is a unit in $E^2(S^2) = E_0$. Since the map from a ring into a localisation of the ring carries units to units, the same will hold for any localisation of $E$.
This procedure doesn't in some sense make the cohomology theory more interesting, but rather less interesting, as localisation usually kills some of the difficulties in the algebra of the original cohomology theory. But that in itself is perhaps interesting.