Both volume computation and lattice point enumeration of convex polyhedron is $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?