The premise of dark matter is that it is something with gravity that does not otherwise interact with any of our detectors (e.g. "normal" matter); or that any such interactions are weak enough that it is plausible we have not noticed.
By itself, that does not exclude the possibility of dark matter having other interactions which do not interfere with our detectors.
I guess so. If you could sort of measure the dark matter distribution, there might be a structure under it, but strictly speaking isn’t it limited to saying it’s not at least likely to emit light but does have mass.
It's very hard to explain the gravitational halo around the galaxies if your dark matter can interact with itself. If it interacted like normal matter, it would have a distribution similar to the gases, and not spread way into intergalactic space.
But it does bound it: if dark matter has self-interactions then the apparent distribution would be different. Regular matter forms stars and galaxies and all the structures we see because it can self-interact.
Since where we see dark matter mass shadows we don't see structure formation, what self interaction it may have must be very, very limited.
Our observations are much more consistent with a form of dark matter which does not self-interact, or which does so incredibly weakly.
Don’t ask me how or why, but this is essentially the universal (pun intended) consensus amongst cosmologists.