If God is omnipotent, does it necessarily follow that He has the ability to not exist? And if God has the ability to not exist, does it logically follow that He does not exist necessarily? But if He does not have the ability to not exist, then He is not omnipotent. And if God does not exist necessarily, then logically God is contingent and not God? But if God cannot not exist, then He does not possess free will. Also, if God is omniscient, does He possess knowledge of reality if He did not exist?
Unfortunately, many use such words as omniscient and omnipotent without understand in detail the meaning of these words in the context of the God of the Bible.
Is God omnipotent? Well, that depends on your definition. There is an unlimited number of things God is not able to do. He is not able to commit an unloving act. He is not able to commit an unjust act. He is not able to commit an unholy act. He is not able to create a rock he cannot lift (one of the common silly criticisms of God’s omnipotence). He cannot do things which violate his nature. A major aspect of God’s qualities is his self-existence (“Before Abraham was born, I AM” John 8:58 “Tell them I AM has sent you…” Exodus 3:14) It is not possible for God to not exist. Existence is necessary as one of God’s qualities. This is not true of us, of course. We exist provisionally, by God’s will. But we are, obviously, not God.
One of the amazing things about God is that his qualities are combined in a supernatural way. His love and his justice could be at cross-purposes, yet at the cross, his love and his justice met, and through the cross God is both just and the one who justifies those who have faith in Christ Jesus (Romans 3:25-26). God’s justice and love do not contradict. God is sovereign, yet his love caused him to provisionally grant us as humans a limited sovereignty which we call free will. God’s sovereignty, therefore, became consistent with his love. Similarly, God’s sovereignty (you are calling it his free will, but I prefer the term sovereignty) is not in conflict with the necessity of his existence. How does the necessity of his existence conflict with his sovereignty?
You are asking what appears to me to be a nonsense question, but that is my opinion, and I hope I can say that without giving offense. Can God conceive of a way he could be just and not loving? Can God conceive of a way he could be both holy and unjust? Can he conceive of a way he can be both omniscient and not exist? Can God conceive of being what he is not? I simply do not know how to answer such a question. Can you conceive of what it would be like if you were square shaped? Can you conceive of what it would be like if time did not exist? This is kind of like the question “Can God create a rock he cannot lift?” This is a nonsense question. Asking what it would be like if a logically impossible thing were to exist is, in my opinion, not a good use of our mental time and energy.
Hopefully, you are receiving this answer in a spirit of having fun with the question, not of admonishing you for asking the question.