Does dark matter follow all principles of regular physics? Is dark matter bound by all the laws of regular physics?
i.e. laws of thermodynamics, speed of light, length contraction, mass-energy relation.
What about Newton's laws of motion (since all of Newton's laws assume an interaction between particles)?
 A: 
Is dark matter bound by all laws of regular physics? i.e. laws of thermodynamics, speed of light, length contraction, mass-energy relation.

Yes. The four things you mention are assumed to apply to dark matter. It does not need to violate any known laws of physics. The only way in which dark matter needs to differ from regular matter is that it doesn’t interact electromagnetically, or it has electromagnetic interactions that are so weak that they can’t be observed. It definitely needs to interact gravitationally. It could have weak and strong nuclear interactions, and perhaps new interactions that regular matter doesn’t have.
It doesn’t need to be weird. All it needs to be is dark, not emitting or absorbing an observable amount of light or any other electromagnetic radiation.

What about newton's laws of motion (since all of Netwon's law assume an interaction between particles)?

Yes. But Newton’s Laws do not assume or require any particular interaction, or even any interaction at all. The First Law tells you what happens when there are no forces. The Second and Third tell you what happens if there are forces.
A: If dark matter consists of particles it would follow all the rules of regular physics.
If dark matter is caused by entropic gravity though (also called emergent gravity; see for example this discussion about Erik Verlinde's theory) dark matter doesn't consist out of dark particles. I think there is proof that contradicts it, but suppose it's true.
In this case, it might be obvious to you that dark matter will not conform to regular physics because of the simple fact it isn't a regular theory from which it originates.
For example, the dark matter in Verlinde's theory will stay inside the blob of matter following upon two blobs of matter that collided (e.g. the bullet cluster) while in the case of dark matter being particles, two blobs of dark matter would appear on both sides of the normal matter blob.
A: 
Is dark matter bound by all the laws of regular physics?

We call "physics" our mathematical modelling of the world's behavior, so dark matter will be covered by its laws eventually.  Its behavior diverges from the behavior we manage to observe with great accuracy for matter in much closer vicinity to us, so either dark matter consists of different particles than what we are used to, or we are overspecialising our theories about matter in our vicinity and overlooking a variable that appears fixed to us, in a similar vein to how there does not actually exist a discrepancy between non-relativistic and relativistic particles.
In that case, the problem is not that dark matter is not covered by what you call the laws of "regular" physics, but that no matter is.  It's just that dark matter gives us the "Doh." about it much less subtly than matter closer to us does.
