It would probably be more proper to say that the galaxy rotates with the dark matter halo, since the mass of the halo is greater than the baryonic mass of the galaxy that we observe. Generally, dark matter halos have triaxial shapes, and shortest axis of the visible part of the galaxy will also be shortest axis of the dark matter halo.
EDIT: I should have been rather more careful. My answer is generally correct, I think, but there may be some more variation than I thought. A paper looking at spin of dark matter halos in the presence of baryons in the Millennium simulation sees a median misalignment between halo and baryonic (visible) galaxy of 30 degrees, and quite a total distribution. They further note that this misalignment seen in simulations will complicate mapping of dark matter halos with lensing measurements.
Yes, dark matter has to be in motion, otherwise it would fall in to the galactic center. From the fact that galaxies are stable, we can expect the Virial Theorem to hold, i.e. that dark matter has a total kinetic energy of half the total gravitational potential of the galaxy.
Yes, indeed, a particular density distribution is required to result in the observed rotation curve; taken the other way around, measurements of galactic rotation curves are measurements of the dark matter density profile. For simplicity (which turns out to be a good approximation) let's assume a spherically symmetric distribution and equate centripetal and gravitational forces:
\begin{equation}
\frac{mv^2}{r}= \frac{GM(r)m}{r^2}
\end{equation}
with $M(r)=\int \varrho(r) 4\pi r^2 \mathrm{d}r$ the dark matter mass profile. Sanity check: Outside the mass distribution, $M(r)$ can be approximated as a point mass $M$ at $r=0$ and one recovers Kepler's law $v(r)\propto 1/\sqrt{r}$, whereas close to the center, $\varrho(r)\sim const$ and thus $v(r)\propto r$. Good. Now, to get the observed flat rotation curve $v(r)=const$ requires $M(r)\propto r$. Such a mass distribution is what you get for a isothermal sphere, often used as the simplest example of a star in ASTR101 courses. A more detailed analysis and in particular a plethora of studies on intricate n-body simulations favors a modification to that profile called the Navarro–Frenk–White profile.
The take home message is that the dark matter velocity distribution is expected to roughly follow a thermal profile, i.e. a Maxwell-Boltzmann distribution. Modifications come from cropping that at the escape velocity, and from a decade-long debate whether the cores of galactic dark matter profiles are "cored" or "cusped". So, yes, our standard models of dark matter phase space distributions correctly reproduce the observed rotation curves of galaxies. Research is ongoing to investigate feedback mechanisms between the baryonic disk and the dark matter halo, and their impact on the observed distributions and scaling relations.
I can provide some pointers if you want, but to get an idea, have a look at the homepage of the Illustris collaboration. Every galaxy you see there is actually a simulated one, so we know the simulated dark matter profile and can compare this virtual universe to the real one to further our understanding of what is going on, despite not yet having detected dark matter quanta directly. The page also lists a number of papers with details for the so inclined. Or, if you prefer plots, from this first to find though somewhat dated paper comes this example of the velocity distribution in a simulated (dark matter-only) galaxy:
(Maxwell is dot-dashed, dashed and dotted other analytical models, solid black is the simulated profile. Green is what this particular paper propagates as a model, purple an idea about the spread seen in simulations. Also note the inset: as promised, the isothermal halo is a pretty good approximation.
Finally,
Have models been investigated for different motion e.g. where the dark
matter orbits at constant radius, or is stationary, or is even moving
at constant speed towards the galactic nucleus, to be periodically
ejected?
As mentioned dark matter can not be stationary in the galaxy's gravitational potential. The dark matter halo is not expected to condense into a disk, because (a) that would require efficient mechanism for dark matter to dissipate energy and (b) disk-only galaxies aren't stable as already some of the earliest n-body simulations confirmed. Yes, in the thermal halo, the individual dark matter quanta are expected to move on elliptical orbits. Ejection (of notable quantities) would mean evaporation of the dark matter halo which is in contrast to the observation that galaxies are around all the time (i.e. are stable)
Best Answer
To answer your two questions:
Almost by definition, dark matter does not interact with itself or other matter at all (or only very weakly). It therefore does not dissipate its energy as, for instance, electromagnetic radiation. "Normal" matter is able to dissipate kinetic energy and as a result can fall deeper into a potential well.
Yes, dark matter is extremely diffuse. Its effects are only felt on very large length scales. The dark matter that exists beyond some particular galactic radius indeed has almost no effect on the rotation of matter inside that radius (it has some, because it not likely to be exactly spherically symmetric). The point is that spiral galaxy rotation curves stay flat out to the edge of where the visible matter is, despite a decline in the visible matter density. The amount of visible mass integrated out to those radii is insufficient to explain the centripetal acceleration observed. The discrepancy can be explained by postulating dark matter that exists inside that radius. However, this dark matter is the minority of the dark matter in a galaxy, most of which is thought to exist in galactic halos and which only (greatly) affects the dynamics of the most distant orbiting objects or satellite galaxies.
The point is made well by this plot from Klypin et al. (2001), which demonstrates how the various components contribute to the Milky Way rotation curve as a function of Galactic radius. Note how the disk+bulge (normal matter) dominate the dark matter (halo) contribution until radii greater than 13 kpc, which is about 4 times the exponential radial density decay scale-length for the Milky Way disk.