Cloth-covered NM questions
The conductors and the insulation inside the jacket are almost certainly perfectly okay.

The only worrisome thing that comes to mind is that the ground wires in those old NM cables, when they had ground wires, were generally undersized relative to the primary conductors.

The electrical code requires that you bring anything you touch during a remodel up to the current code.

The current code basically requires that the ground wires be the same gauge as the primary conductors (#14 for 15A circuit, #12 for 20A circuit, etc).

I don't know if there's an exception for this old cable.