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, journal) 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.