1) Convergence will be easier to achieve and the solution will take less time to compute if you use two orthogonal planes at the center and slice the sample body in 4 pieces and only simulate 1/4 of the sample, using two Symmetry Regions on the two cutting planes. Note that if your sandwich has honeycomb shaped cell walls, you must position the center cutting plane at a symmetric part of the cell, like at the middle of a cell.

2) The rods should all be Rigid bodies which means only the outside surface of the rod will be meshed.  You don't need to apply symmetry to the rod bodies. The Remote Displacement boundary condition on each rigid body will control all 6 DOF.