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A counterexample to for both the first and third questions can be found in Counterexamples in Topological Vector Spaces by Khaleelulla (p. 108).

Let $W$ denote the space of all $\mathbb C$-valued sequences $(x_n)$ and $\Phi$ the space of finite sequences. Let $E=E_1\oplus E_2$ where $E_1$ is the topological countable direct sum of copies of $W$ and $E_2$ is the topological countable product of copies of $\Phi$. $E$ is a complete locally convex space but the quotient $E/M$ where $M=\{(u,u):\ u\in E_1\cap E_2 \}$ is not even sequentially complete.
show/hide this revision's text 2 added 124 characters in body; deleted 4 characters in body

A counterexample to both the first and third questions can be found in Counterexamples in Topological Vector Spaces by Khaleelulla (p. 108).

Let $W$ denote the space of all $\mathbb C$-valued sequences $x = (x_n)$ (x_n)$ and $\Phi$ the space of finite sequences. Let $E=E_1\oplus E_2$ where $E_1$ is the topological countable direct sum of copies of $W$ and $E_2$ is the topological countable product of copies of $\Phi$. $E$ is a complete locally convex space but the quotient $E/M$ where $M=\{(x,x):\ x\in M=\{(u,u):\ u\in E_1\cap E_2 \}$ is not even sequentially complete.
show/hide this revision's text 1

A counterexample to both the first and third questions can be found in Counterexamples in Topological Vector Spaces by Khaleelulla (p. 108).

Let $W$ denote the space of all $\mathbb C$-valued sequences $x = (x_n)$ and $\Phi$ the space of finite sequences. Let $E=E_1\oplus E_2$ where $E_1$ is the topological countable direct sum of copies of $W$ and $E_2$ is the topological countable product of copies of $\Phi$. $E$ is a complete locally convex space but the quotient $E/M$ where $M=\{(x,x):\ x\in E_1\cap E_2 \}$ is not even sequentially complete.