Wouldn't it -very- slowly lose energy due to gravitational radiation?

If the cloud was inhomogeneous, yes. However, (unless I am mistaken, which is fairly likely,) the more symmetric a bulk collection of particles (such as WIMPs), the less gravitational radiation it would emit as it oscillated. In particular, for every particle falling in from one side of the planet, you have another particle falling in from the other side of the planet, so the mass distribution of the system does not change over time. I don't know how gravetomagnetic effects would come into play.

> speed past the center

What caused these particles to accelerate up to the speeds they are at in the first place, then? Something must be interacting with them quite significantly if they have non-zero velocity?

Gravity.

Actually, almost the only thing that we know about dark mater is that gravity affect it.

(Actually, we only know that there is something that is making a gravitational pull of normal mater, and we call this mysterious thing "dark mater".)

the force of gravity

Aha, DM particles are in eternal orbits!

Thanks!