It is possible for an object to be in the shadow of Earth permanently - no special restrictions on the Earth or Sun needed.  There is a special place called the [L2 Lagrange point][1] that allows an object to orbit around the Earth at about the same rate as the Earth orbits around the Sun - thus always remaining in shadow.  

However, L2 is not a super stable point to orbit in, and thus objects tend to fall away. This is not a problem for artificial satellites, though, and the [James Webb Space Telescope][2], when launched, will take advantage of this constant shadow to make better celestial observations.

So: 

1) Yes, it is possible, although unlikely.  This "moon" is more like the Death Star.  Will need some kind of course correction.

2) Current Earth / Sun distances, and the Earth / Moon distance would be about 1.5 million kilometers out, which is about 3-5 times the current Earth / Moon distance.

3) So, the Lagrange point kind of orbits both.

4) Normal densities (3,000-8,000 kg/m^3) should work.  However, some kind of automatic correction will be necessary.  



  [1]: https://en.wikipedia.org/wiki/Lagrange_point#Lagrange_points
  [2]: https://en.wikipedia.org/wiki/James_Webb_Space_Telescope