I believe the grain orientation evolution in DAMASK is calculated based on the rotation component of the elastic deformation gradient (Fe); and, before the stress divergence calculation starts, the lab and the crystal coordinates are aligned using the rotation matrices from the initial grain orientations. So, basically, you pre-multiply the rotation matrix (from your initial crystal orientation) such that the lab and the crystal frames are now aligned, perform the stress-divergence calculations to get the correct/consistent Fp and Fe, get the updated rotation from the Fe, and finally post-multiply with the rotation tensor (from the initial crystal orientations) to be consistent with your starting frame.
In short, you don't have to worry about rotations of the elasticity tensor, as every thing is done in a consistent way internally in the code itself during the run.
Yes, each material point in the simulation keeps (internally) track of the respective orientation. The specific way of how we do this in DAMASK ensures that in the intermediate configuration the crystal frame and lab frame are always aligned. Hence, only one (non-rotated) version of the elasticity tensor is needed to calculate the stress (based on elastic deformation) at any point in the geometry.