Open mapping theorem for complete non-metrizable spaces?

Question 1. Such results have been studied in detail—-a good reference is Köthe‘s monograph on topological linear spaces. You could also look up the concept of webbed spaces (de Wilde).

For question 2, you can take the space of bounded, continuous functions on the real line—-it has two distinct complete locally convex structures: that induced by the supremum norm and the strict topology which was introduced by R.C. Buck in the fifties.

Another example for question 2 is the following: There are linear partial differential operators with constant coefficients (e.g., the wave operator) and open sets $\Omega \subseteq \mathbb R^3$ (e.g., the complement of a cone) such that $P(\partial)$ is surjective on the space $\mathscr E(\Omega)$ of smooth functions but not on the space $\mathscr D'(\Omega)$ of distributions. Then the range $X$ of the transposed $P(-\partial):\mathscr D(\Omega)\to \mathscr D(\Omega)$ is a closed subspace of the space $\mathscr D(\Omega)$ of test functions and it has two different complete locally convex topologies: The subspace topology from $\mathscr D(\Omega)$ and the strictly finer topology making $P(-\partial): \mathscr D(\Omega) \to X$ a topological isomorphism.