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.
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 \}$is not even sequentially complete. 1 A counterexample to both the first and third questions can be found in Counterexamples in Topological Vector Spaces by Khaleelulla (p. 108). Let$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.