$\def\sp{\kern.3mm}\def\TT{T}$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.

$\def\sp{\kern.3mm}\def\TT{{\mathscr T}}$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 identifying $X$ 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 $r<s$ and $(r\sp,s)\not=(2\sp,3)$ the identity 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.