Wow I see the light (I think) :P
I'll just take a quick guess here, and say that a small addition to your answer to my second question would be 'only if your element is a point mass, or the element has moment of inertia = dm(r^2) ' ?
I think this follows, looking at the working for a solid cylinder, my textbook uses shells, which each have moment of inertia dm(r^2), and that derivation just uses I = integral of r^2 dm, which is the same working for a point mass element as you point out.
Anyhow, I'll go and review this stuff. Thank you for your explanation!