Thanks for this nice comment. I am really trying hard to make it easy to understand, but as I am biased, I am easily screwing it up (this already happened in the past on this very same platform) ;)

How is dark matter distributed in a galaxy? Is it more like homogenous throughout, at the periphery or it has its own local clusters like normal matter? Is there any hint through velocities of stars in the galaxy that can reveal the distribution?

Dark matter is by far not homogeneously distributed. First, dark matter tends to be where normal matter is. The reason lies in galaxy formation: in order for the galaxies to be what they are today, dark matter seeds were needed in the early days of the universe so that gravity could act and make galaxies forming. Locally, it is assumed that there are dark matter halos around the galaxies, as evidenced by the galaxy rotation curves (otherwise, the stars lying far from the galactic center would rotate not as fast as observed). But the halo itself is not uniform and more on the clumpy side.

1. I assume muonophilic dark matter as of now is a hypothesis. How did we come to this conclusion, that some dark matter may love muons but not other forms of normal matter. Or do we also have electronphilic, heavy quarkphilic dark matter too? What made muon so special?

It is indeed a new idea. As dark matter has not been directly observed, physicists have to think how this is possible. And this is how new ideas like muonphilic dark matter pop up. This may be one good reason why dark matter has not been observed so far.

I have myself worked on topphilic dark matter (strong connection to heavy quarks), see for instance here. I do not remember having reads anything about electronphilic dark matter, but there are a few papers on that too, especially when dark photons are also involved (I should however read them to give you more insights).

I hope it clarifies. I have to leave now so that I can't answer more longly, but feel free to come back to me for more precisions!