That doesn't rule out "dark matter having other interactions which do not interfere with our detectors" Dark sector theories which include other dark particles or new particle interactions are not controversial amongst cosmologists in this space. For example there's a whole area of study around "dark photons" which would mix with our photons and interact with dark matter.
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.