These are my thoughts.
1. Is a lie a sin? Yes, a lie is a sin. If God cannot lie (Tts 1:2), then it is not a stretch to say He cannot sin.
2. If you adhere to Reformed Theology, we know that we can only choose according to our nature. So God is free to choose only according to His nature.
Some other thoughts to consider. God is such a holy and loving God, and for Him to be able to sin would mean that He would fall under his own wrath. His justice then would not preclude Him from such a disastrous end. Since "God is light, and in him is no darkness at all (1 Jo 1:5), this strongly suggests that there is not so much as a quark, a small seed of sin in Him, if there were, He would be sin Himself. For sin is expressed in its full blown hatred and evil contemptuousness, in JUST ONE lonely violation of the law, that's all it takes.
All sin is in rebellion against a holy and righteous God, why would God be at odds with His own nature, it would be self-defeating, and cancel out His very nature? God could then not be LOVE, for He would have to hate Himself. Also, if God were able to sin, what would prevent Him from somewhere in eternity turning evil? On what grounds or basis would I be able to put my trust if this were the case? The Anti-Christ has this ability, appearing to be an angel of light then turning on his foes to seek, kill and destroy.
Also, Adam was born is a state of moral equipoise, he had no particular bias as to sin or not to sin. His nature was NOT in a fixed state of holiness, so therefore his ability to sin existed. Adam was created in a state of innocency, and not in a state of holiness.
Remember A&E did not fall after eating the apple, there was sin present in their heart to begin with, in order for them to WANT to disobey God.