No. The effects of dark matter on galaxies, for example, could potentially be explained by a modification of the gravitational force. In this theory, “dark matter” never existed in the first place.
Maybe – there are theories of an extended “dark sector”, which involves, for example, “dark electromagnetism”. In dark electromagnetism, dark matter can interact with other dark matter via “dark photon”. There is a variety of tests one can conduct to deduce whether dark electromagnetism (or any other “dark forces”) is true or not; so far the tests are inconclusive.
We do not know. Different models of dark matter predict different “sizes” for the dark matter particle.
-) At small scales (i.e., solar system and smaller): we have no idea, again different theories predict different densities
-) At large scales, this can be measured using techniques like “gravitational lensing”, where the gravity of the dark matter bends light that we can observe with telescopes
-) At extremely large, cosmological scales, the prevailing theory is that it has a density of 0.23 times the “critical density”, where the “critical density” is density required to make the Universe flat (it’s just a number that you can compute)
Yes, it’s difficult, but we’re trying our best.