Dark matter and gravitation

Generally, it is difficult for dark matter to clump. That's due to angular momentum conservation: Normal matter only clumps so efficiently into stars, planets and all that because we have are dissipative forces ("friction") that permit efficient clustering.

For dark matter on the other hand, we do not expect any (or at least no strong) dissipative forces; this is the Lambda-Cold-Dark-Matter paradigm where dark matter only interacts through gravity. In that case, clustering can only happen through three-body interactions (known more readily from swing-by maneuvers of solar system probes, where one body gains angular momentum at the expense of another loosing it). And in the vanilla scenario that means we don't expect any (or at least not much) structure at scales much smaller than dwarf galaxies: no dark matter stars or planets expected.

That said, there are proposed modifications to this vanilla cold dark matter scenario that are consistent with data, in which the dark sector has extra forces, which in turn could induce stronger clumping. Self-interacting dark matter is one example of one such model, see e.g. this nicely written though somewhat outdated paper.

Dark matter is matter too, so yes, any motion of a dark matter clump gives the same gravitational structure (and potentially also gravitational waves) as a clump of the same size and mass that was made of normal matter. But if one had different density profiles and very sensitive gravitational wave detectors one might tease out a different signature.

Thinking purely phenomenologically, dark matter quanta could be anything from 10^-22 electronvolts light up to 10 solar masses heavy (roughly. Let's not go into arguing about those exact limitations in this post). At this massive end of the allowed range, primordial black holes are one candidate that is being discussed in the literature. In fact, this model recently got a lot of attention exactly because of the early gravitational wave signatures seen by aLIGO, though by now it is disfavored as making up all of the dark matter.

tl;dr: No, we do not really know if there are any star-sized clumps of dark matter, yes we could see them in gravitational waves.