I believe it's also reasoned to be a consequence of the apparent non-interaction with the electromagnetic force that dark matter has. Since it doesn't block light in any way we can detect (absorption, changes in polarization, changes in direction, etc.) it's also not likely to actually "touch" normal matter in the way you'd intuitively expect. That's because that touch is mediated via the electromagnetic force, the electron clouds and the charges they have repelling each other over the vast empty spaces of normal matter.

Strictly speaking we don't, but we have reasons to believe that on a galactic scale it doesn't. There was that famous example a few years back where we had observed two galaxies hitting each other, and it "looked" like the regular ol' matter (eg, interstellar gas clouds) in each galaxy had "hit" each other and stopped while the dark matter in each had passed through the other galaxy unimpeded.

Which would seem to indicate that either dark matter is sparse and heavy (and just doesn't hit things very often) or diffuse and not-very-interacting with other matter.

I think they mean we only observe other matter interacting with other matter, and not dark matter interacting with other matter.