Can God kill Himself.?" This presumes God is a physical and material being. I’m afraid I don’t see why being non-physical entails being eternal. For example, couldn’t God create an angel and then destroy it later? If angels are non-physical beings that can be created and destroyed, then immateriality doesn’t entail eternality. Moreover, you’re right that God cannot die, but it doesn’t follow that the answer to question #1 is “no”. If there was something that God couldn’t do, then God wouldn’t be omnipotent. So the question asks can God commit a logically contradictory action.

God would then be both a non material being, and a material being in which he animates, that has the potential to lift the stone. Now if you belive that every material object has consciousness… I think our starting assumptions are somewhat far apart.