Is there any evidence that dark matter interacts with ordinary matter non-gravitationally? As far as I understand it, dark matter direct detection experiments are based on the idea that dark matter interacts with ordinary matter, it just has a very small cross-section. So far, there's been no confirmed detection.
Is there any evidence at all that dark matter interacts non-gravitationally with ordinary matter at all? That is, is it possible that dark matter exists, but can literally never be detected because every non-gravity interaction cross-section is exactly zero? 
If there is evidence, what is it? If there is no evidence, why does this hypothesis seem never to be taken seriously?
Related: Is there any evidence for dark matter besides gravitational effects? and Why do physicists assume that dark matter is weakly interacting? However, neither question deals explicitly with the hypothesis that dark matter doesn't interact non-gravitationally.
 A: There are some standing anomalies that could be explained by non-gravitational dark matter interactions. For example, Fermi-LAT is an indirect detection experiment (i.e. an experiment that looks for the debris of a dark matter decay that occurred far from Earth), and it currently reports an excess of gamma rays. There are occasional claims that nontrivial dark matter interactions make galaxy simulations work better. Excess cooling of astrophysical objects, such as white dwarfs, also "hints" at dark matter which could carry extra energy away. Finally, there are even some direct detection experiments like DAMA which claims to see dark matter interactions. 
However, these anomalies generally don't point in any coherent direction, and often fade away. Most of the time, all they reflect is an incomplete understanding of astrophysics. So you're completely right: the default hypothesis, if you're an astrophysicist, is that dark matter has no non-gravitational interactions.
But particle physicists are investigating a different problem: they're trying to explain what dark matter is made of and how it got here. This always takes place within specific models. No direct detection experiment just looks for "dark matter" per se; instead they look for specific dark matter candidates. 
The longest-running direct detection experiments look for particles whose mass is about a hundred times the proton's, and whose interactions are primarily electroweak. This specific guess is because such dark matter candidates exist in many supersymmetric extensions of the Standard Model, which were favored for other reasons. In such models you automatically get dark matter interactions; you can't get rid of them unless you purposefully tune the interaction to zero. Moreover, when you run the numbers you find that such models automatically produce the right amount of dark matter in the early universe, a result known as the WIMP miracle. (Production is another reason you want interactions; you could just postulate you get the right amount by tweaking the initial conditions, but it would be nicer to explain why.)
These days, a wider range of dark matter candidates are getting attention, leading to a great variety of cheap direct detection experiments, which target each candidate's particular interactions. Everybody working in this field (which comprises less than 1% of physics at large) is perfectly aware that all these candidates could be nonexistent, and that the whole enterprise might have been completely doomed from the start. But that is true for any scientific endeavour; that risk is what makes it exciting in the first place!
