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: 
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.
Adding two points that havent' been made here yet:

* I would argue that observations of Baryon-Acoustic Oscillations in the power spectrum of the Cosmic Microwave Background are actually the strongest evidence that the universe contains a large amount of matter that clumps under gravity but does not feel any photon pressure.
* Since dark matter doesn't interact via electromagnetic radiation, its name is a misnomer. It should be called transparent matter instead.