The answer to 2 is "No". Observe that a positive answer here would imply all Delaunay simplices to be unimodular (i.e., have volume equal to $1/n!$ times the volume of a fundamental parallelepiped). This holds for $n\le 4$ but starts to fail for $n\ge 5$. See, for example, my paper "Lattice Delone simplices with super-exponential volume" ([arXiv][1], [journal][2]) and the references therein.

I am not sure about the other two questions; I would be very surprised if 3 is true, and have no clear opinion on 1.

[1]: http://arxiv.org/abs/math.CO/0507119
[2]: http://dx.doi.org/10.1016/j.ejc.2005.12.003