What tool is a solid metal cube with dimples or half spheres on every side

For thicker pieces you might find you get better results by compressing the foam before cutting it. You could make a simple straight-line clamp out of a couple of 2x4s linked with a door hinge -- just put your foam in between, clamp the open end shut with a C-clamp, and use one edge of the wood as a guide for your cut. For your curved cuts it might make more sense to make a plywood die to match the final shape and press it down by hand or with something heavy before cutting around it.

You can use a Dremel cutoff wheel to cut off a piece of the rod but you may have to remove the rod to do this. You can use Dremel grinding wheels to grind down the end of the rod. If the rod is not protruding out of the hole, you could use a wooden plug to seal off the hole. Sand the plug flush with the bench top.