Question: How do we know that dark matter is dark, in the sense that it doesn't give out any light or absorb any? It is impossible for humans to be watching every single wavelength. For example, what about wavelengths that are too big to detect on Earth?

Answer: 
If dark matter emitted very long wave lengths of electromagnetic radiation it would mean it is composed of charged particles. There is no escape from that conclusion. Somebody might propose that dark matter is some strange configuration of charged particles which acts as a very long wavelength antenna. That might be a good model, but there is a hitch with it. If it emits longer wavelength radiation it must be colder. The Wien displacement law is that $\lambda~=~c/T$, for $c$ a constant, which may be calculated, but is not relevant here. For $T~=~2.7$K the wavelength of radiation is $.1cm$. Let me assume that there does exist some very long wavelength radiation, say $100m$ as the lower bound on this. The ratio of temperatures gives that
$$
T\_{100m}~=~\frac{10^{-3}}{10^2}2.7K~=~2.7\times 10^{-5}K 
$$
So all this can be is a very cold gas, which is the hitch. This gas is much colder than the background radiation and not in equilibrium. So from some physical grounds this is not likely, and dark matter is most likely not some ordinary form of matter that interacts by EM.
There is indeed very good reason to believe dark matter is dark - apart from all the evidence from "missing mass" in luminosity counts and gravitational lensing studies.

This comes from theories of large-scale structure formation:
That there has always had to be some sort of matter that doesn't interact electromagnetically at all is crucial to most scenarios of large-scale structure formation. The density fluctuations in the present universe would be too large than what would be predicted if there were only ordinary baryonic matter that interacted only electromagnetically. With dark matter, you can have something that gravitates yet decouples from radiation much before baryonic matter does. This allows the dark matter to form gravitational wells (under collapse) which have a much longer time to expand with the universe. By the time ordinary matter decouples from radiation and joins the rest of the expansion flow, the ordinary matter will quickly fall into these large gravitational wells of the dark matter that have had far more time to grow. This, in a way, amplifies density perturbations in the early universe and allows large-scale structure to form. (to the extent that we see it today in the form of clusters and galaxies)

The required amount of dark matter calculated, in this way, in order to observe the present scale of density fluctuations matches very well with the amount of dark matter required to explain galactic rotation curves, gravitational lensing, etc. So there's excellent agreement that all of these are due to the same thing - some sort of matter which doesn't interact electromagnetically at all, viz. dark matter.