If dark matter can't lose kinetic energy, then why is it not traveling at relativistic speeds? I have read this question:

The only way you can do this is to remove kinetic energy from the system.
With normal matter this is done through electromagnetic interactions, which turn the kinetic energy of normal matter (protons, electrons etc.) into photons, which then escape from the system. Since these kinds of interactions do not occur for dark matter (by definition), then there is no way to get rid of kinetic energy and so the dark matter remains as a large "halo" around gravitationally clumping ordinary matter.

If dark matter only interacts with gravity, why doesn't it all clump together in a single point?
And this one:

A significant portion of the dark matter is known to be gravitationally bound to galaxies and relativistic velocities are far about the escape velocity and the stuff you propose would not remain bound.

Can dark matter be relativistic dust?

I assume that its speed with respect to the Sun will have a distribution with an rms of a few 100 km/s.

Can the Sun capture dark matter gravitationally?

In general particles created at the time of Big Bang are at non relativistic velocities at present .

What is meant by dark matter being non-relativistic and why is this?
The first one says that dark matter cannot lose kinetic energy through interactions.
As far as I understand, most ordinary matter moves at non-relativistic speeds around us because these are able to lose kinetic energy through interactions and thus we see objects made of these particles in the observable universe to be mostly non-relativistic (except for example neutrinos).
Now if dark matter cannot lose kinetic energy, then shouldn't it have most of its kinetic energy from the Big Bang and travel at relativistic speeds? The second one says dark matter is definitely non-relativistic.
Question:
If dark matter can't lose kinetic energy, then why is it not traveling at relativistic speeds?
 A: It loses energy as the universe expands, in the same way that a gas cools as it expands. The same happened to the cosmic microwave background radiation (CMB), which is why its effective temperature is only a few Kelvin, even though the universe was about 3000 K when this light was first emitted during the Big Bang (at a moment called 'recombination').
To give a slightly more detailed answer, in an expanding universe, everything is pushed away from everything else, and the further away two objects are, the faster they move apart. This tends to reduce the relative velocities of particles. For example, if you surround yourself in a sphere of dust particles, they will gradually accelerate away from you. Thus any particles moving toward you will slow down (and it's the particles moving toward you that you will measure when you measure temperature).
A: If dark matter consists of particles that thermally decoupled from the rest of the universe very early, then its momentum distribution with respect to the comoving reference frame was fixed at that point.
As the universe expands, then the characteristic absolute scale of this momentum distribution decreases as the scale length grows.
One way of thinking about this is that the de Broglie wavelength of a particle $\lambda = h/p$, gets stretched by the universal expansion just like the wavelength of light. Hence the rms $p$ (with respect to the comoving reference frame) decreases and the particles become non-relativistic.
A: Now that's an interesting question. And truth be told, as with so many things about the characteristics of dark matter, we do not know the answer yet.
First, it is not correct that "dark matter can't lose kinetic energy". We wouldn't have galaxies if that were the case. A population of dark matter can cool through gravitational three-body interactions. You know those from space probes making swing-bys on planets, typically to increase their kinetic energy, or if you travel to the inner planets, also to shed kinetic energy, at the expense or gain of the third body's kinetic energy. This is how dark matter halos cool and thus clump, which in turn stabilizes the formation of the baryonic disks that we have all come to love so dearly. This process is not as efficient in shedding kinetic energy as e.g. friction, so dark matter halos are very puffy and we should not expect dark matter discs or dark matter stars, but nonetheless, the process exists.
Indeed, as you say, from structure formation we know that dark matter is non-relativistic (aka "cold") and thus explains the structures we observe in the universe. Your question is why that is the case, and we don't know, because we don't know how dark matter came to be. Usually this is taken the other way around: we know dark matter is non-relativistic, and we use this to weed out all kinds of models and dark matter production mechanisms that are in disagreement with that.
The two most popular production mechanisms that produce non-relativistic dark matter are massive "thermal relic" particles and the "misalignment mechanism". Thermal relics are any particles that are at some point in the past in equilibrium with the rest of the universe's plasma. Protons and neutrons are a known example. WIMP dark matter is the most famous hypothetical example. By definition, thermal relics have a Maxwell-Boltzmann velocity distribution at some point. As the universe cools, that gets redshifted, thus leaving you with non-relativistic dark matter. Again, protons and neutrons do the same thing. This works well for dark matter that is more massive that a few proton masses. In turn, this rules out e.g. neutrinos as a dark matter candidate, as they are too light and would move relativistically.
The other example mechanism, misalignment, is invoked to explain why axion dark matter would not be relativistic. Axions are extremely light, but if your model prevents them from ever being in thermal equilibrium (because they interact too feebly), then their velocity distribution is set by their production mechanism. If you come up with some way to produce them close to their quantum mechanical zero point energy, then that is nonrelativistic and thus matches observation.
In summary, dark matter moving slowly is an observational fact that needs to be explained by any model attempting to describe the nature of dark matter, and indeed this constrains the possibilities for model builders.
