What Gravitational Waves Can Say About Dark Matter (opens in new tab)

Actually dark matter clumps, e.g. so called dark matter halo of a galaxy[1]. It is seen from rotational curves of galaxies and can be also checked with with strong and weak lensing effects [2].

[1] https://en.wikipedia.org/wiki/Dark_matter_halo

[2] https://en.wikipedia.org/wiki/Gravitational_lens

Yes, but this just upper bounds the interaction strength rather than constrains it to be zero. Additionally, if there are multiple species of dark matter, there may be subpopulations (up to 10%, I think) that interact very strongly without conflicting with observation.

In even the classic theory, when the Swartzchild radius of the gravitational wavelet exceeds its width, wouldn’t it naturally form a black hole and end up a cusp?

Like a gravity Kugelblitz.

It is presumed that the dark matter also rotates about the galactic center. It can’t fall in, any more than the stars in the galaxy can, because it collectively can’t dump its angular momentum. However, there is controversy whether dark matter can dissipate momentum that is parallel to the rotational axis either by self interaction or by interaction with ordinary matter. If it can, then the distribution of dark matter could resemble a disk, as opposed to a spherical distribution.

Lisa Randall has written a provocative book about this which will answer your question in depth: Dark Matter and the Dinosaurs: The Astounding Interconnectedness of the Universe.

As others have pointed out, if they’re in a stable orbit they won’t fall in (at least not for a very long time), because unlike luminous matter it won’t experience any force other than gravity. If you imagine a dust cloud of luminous matter around a black hole, it will tend to experience frictional heating the closer it gets, there is the chance of a collision or radioactive decay, and other forces acting to draw it in or send it far away. Dark matter won’t do that, it just couples to gravity. Our usual intuition about how a halo of matter behaves has a lot to do with interactions other than gravity. Clumping for example, aggregation and accretion pretty much work because of interactions other than gravity, until a body becomes massive enough.

They do. But there's nothing special about the gravity of a black hole versus the gravity of "ordinary matter". The vast majority of the matter in a galaxy is in orbit and won't get anywhere near close enough to the central black hole to fall into it. This is true of stars, planets, gas, and dust just as it's true of dark matter.

They can still get rid of momentum in N-body interactions via gravitational waves (N>=2) or by transferring to it to another body (N>2). The question is whether those mechanisms are significant enough to lead to compact bodies.

Dark matter objects interact with regular matter only via gravity. But the article seems to suggest that it might interact with itself by some other force, which may allow for "collision":

"Dark matter seems to interact with normal matter only through gravity, but, based on the way known particles interact, theorists think it’s possible that dark matter might also interact with itself. "

If you did a PhD (Emphasis on PhD because an undergraduate degree isn't close to enough) on particle physics solely with career prospects in mind then you probably have bigger issues