Only a comment on Question 2: It is already not easy to define the von Neumann algebra generated by a family of closable operators, see e.g. https://projecteuclid.org/euclid.cmp/1103899047 Def. 2.5 and Rem. 2.7.

Closability in this context appears naturally for the following reason: Assume that you have some (possibly non-closed) unbounded operator $M$ on some dense domain $\mathcal D$. To get towards defining a corresponding von Neumann algebra $\mathcal M$, I presume you would first try to compute a formal adjoint $M^*$ and I think you will agree that $M^*$ should also be densely defined. But then you already obtain that $M$ is closable.