Could there be such thing as anti-dark matter? If there is anti regular matter, why not anti-dark matter? If there is no such thing as anti-dark matter, why?
Yes, it is a thing.
Weakly Interacting Massive Particles (aka WIMPS) are thought to come in matter and anti-matter forms and have a self-annihilation cross-section in order to produce the "correct" amount of dark matter (in relation to photons) that we measure today.
As the opening paragraph of the wikipedia article on WIMPS states - there are a number of experiments searching for the photons associated with Annihilation of WIMP anti-WIMP pairs, from which it appears that there is expected to be a neutrino and a gamma ray signature or even the presence of anti-matter (positrons) that might be detected in cosmic ray experiments.
Of the other dark matter candidates - whether sterile neutrinos have an anti-particle depends on whether they are Majorana fermions or not. The hypothetical axion I think is its own anti-particle (a chargeless boson) and there are no annihilation-type experiments for their detection that I can find.
It depends on the composition of the dark matter.
From the description I read it is clear that MACHOs (composed of normal baryonic matters) can have anti-objects, and there are also anti-WIMPs.
I am not sure about the Axion, it seem to be a different beast (on par with the photon that is its own anti-particle).