$\def\sp{\kern.3mm}\def\TT{T}$Still further (counter)examples of the situation in Q2 are obtained from Propositions 4.4.3 (p. 81) and 6.6.7 (p. 111) and Example 6.10.L (p. 123) in Jarchow's *Locally Convex Spaces* as follows. Let $F_3=(X,\TT)$ be any infinite-dimensional Banach space, and let $F_r=(X,\TT_r)$ where $\TT_0$ is the finest linear topology for $X$. For $0<r\le 1$ let $\TT_r$ be the finest locally $r$−convex linear topology for $X$, and let $\TT_2$ be the "box topology" for $X$ obtained by indentifying $X$ with the $B$−fold direct sum of the scalar field for any Hamel basis $B$ for $X$. Then taking any $r,s\in[\sp 0\sp,1\sp]\cup\{\sp 2\sp,3\sp\}$ with $r<s$ the indentity is a *continuous linear map $F_r\to F_s$ of complete topological vector spaces that is not open*. For $1\le r$ we even have locally convex spaces that, however, the OP did not require.

For Q1 I recommend to see Theorems 5.5.2 (p. 95) and 11.1.7 (p. 221) *loc.cit.*