> Dark matter can be placed arbitrarily

Yes.

> so it can match pretty much anything

At the time you do the initial placement. But that's not the only relevant time.

You can scatter any type of DM on a spacelike hypersurface any way you want, but then you have to figure out the successor (and predecessor) spacelike hypersurface, and there you have to bite a bullet: you can constrain the DM or throw away the well-posedness of the initial-value problem.

Your constraints are dominated by the hyperbolic-elliptic PDEs. The hyperbolic portion lets you lay down initial values for your dark matter on a spacelike slice. You then need an equation of motion for your DM, as you do for your visible matter, so that you can evolve the contents of that initial spacelike slice in time. The elliptical portion are the Hamiltonian and momentum constraints.

Without satisfying these constraints you tend to lose uniqueness, which destroys the predictive power of your model.

If you're spraying down a dust of non-interacting DM particles within some spacelike region, as you evolve from your initial configuration you end up with each particle growing into an everywhere-timelike worldtube. (Non-(self-)interaction is just to make this really clear without the distractions of decays or daughter products from scatterings. Each dark matter particle has its own worldtube extending arbitrarily far in to the future or past.)

The simplest Cold Dark Matter models have a conventional nonrelativistic and relativistic dynamics determined by two parameters: particle rest-mass and interaction cross-section. In our simple non-interacting cold DM dust case the interaction is purely gravitational, so we're left with every particle having an identical rest mass.

Now, lay down the visible matter in a galaxy (or cluster!). You can lay down the dark matter dust arbitrarily to form a halo which could support a realistic galaxy rotation curve on your initial surface. You are right that you can put it practically anywhere, and are not restricted to uniform density. However, when you evolve in time, you inevitably find that you are stuck with a narrow range of mass densities at various points in the galaxy (or cluster!), but still have freedom to set down differing particle number density, higher numbers at lower particle mass. If your particles are too light, the gravitational coupling with the visible matter is too slight; worse, the gravitational coupling between the DM particles is too slight too, and your galaxy quickly disassembles. If your particles are too heavy, instead you get dynamic heating of hydrogen gas, and you get far too much star formation and metal enrichment.

You can go the other way too: first put down a test particle DM halo and then add hydrogen gas, or a modern galaxy, or whatever else, and see how the system evolves. It is a feature of the standard galaxy astrophysics that DM and visible matter do not have to be near each other at all times. (This gets important when you consider galaxy clusters rather than just galaxies; laying out CDM is hard work, as I've been describing, but you can't just drop in MOND. It's also testable at solar-system scales; we can look for post-Newton dynamics in the Jupiter system, or in the inner solar system, and test whether it's compatible with General Relativity (with a fine dark matter dust of up to a few Phobos-masses inside Neptune's orbit) or whether it's compatible with a different post-Newtonian dynamics, like MOND, where the galaxy's MOND corrections will generate a quadrupole anomaly stronger around the outer planets and their moons than around the inner planets).

More complicated particle CDM theories are still stuck with the general rules of the simple model above: the stress-energy in the large has to hang around in the right places for long periods of time; the stress-energy can't just pop in and out of existence at a point (it needs to be thermal, so its worldtubes cannot even be nearly lightlike, let alone timelike); and it has to have a reasonable number density). Crucially, non-gravitational coupling to the Standard Model imposes further restrictions on the initial surface, it does not relax them.

More complicated particle CDM is usually motivated by problems in particle physics, models for for which galaxy (and cluster!) astrophysics can provide observational evidence. Some WIMP models are good tests for SUSY models, so one experimental search might test two models grown from very different underlying theories. One does not have to believe the "WIMP miracle" conjecture when one can outright test it observationally and in a laboratory setting.

> MOND

By construction you can't put MOND just anywhere on your initial values surface; it is a phenomenon arising from visible matter. You can't put down MOND first and then add matching matter and see what happens. (Well, there are relativistic theories which start by taking MOND and then proceed to take General Relativity by introducing one or more fields until they get a good match with MOND. Some of those approaches might let you put down a "MOND field" first and add matter that -- Famaey and McGauch have a good, if aging, overview in Chapter 7 of https://arxiv.org/abs/1112.3960 -- but at some point you might ask if you can start with an already-relativistic theory like General Relativity and recover MOND from that instead. MOND does appear to work for galaxies, after all, even if it needs modification for much larger and much smaller structures.)