The last half of John 1-3 states, "Apart from Him nothing came into being that has come into being. Does this imply that some things never came into being?" I'm going to stop right there and answer that part of the question. The whole point of that phrase, "apart from Him nothing came into being that has come into being," is not really to make us ask the question, "Are there potentially things God could have created but He didn't?"
There's no way to know that. And I'd rather just simply say that what that tells us is that God, through Christ, is the creator of everything. There isn't a single thing created that came outside of His work. And that obviously speaks to the issue of evolution. It speaks to the issue of those who believe maybe God was partially involved in creation and there's a whole slew of different varieties of belief systems. It's just a way of saying everything that exists is here because of God. And there was another part of the question that says God never came into being. He always was. God didn't have to create logic. It was always there. 2+2=4 was in existence before God began His creating work.
And here's the crux, was sin and evil in existence before eternity passed. First of all, huge volumes have been written on this subject and so for me to try to answer this in just a moment really isn't going to be effective. But we'll say this, God is not a sinner. Scripture is very clear about that. God cannot sin. He cannot touch sin. He is pure. He is light. He is completely holy. He did create His angelic beings with the potential to make a choice and Satan, I don't subscribe to the idea that Lucifer was his name, that was a translation issue, but Satan made that choice and sin came into the world because of his choice. And I'm not one that says that you have to fully grasp every area of theology in order for it to be true. I think we run up against a wall of how is it that God is perfectly holy and pure and He created the potential for sin to enter into the universe and it did, yet He's not touched by it. I don't think we have a good answer to that except the Bible says God is holy, He is pure, He cannot touch evil and He does not touch evil. So that part is I think very clear from Scripture.
But then an interesting part of the question, did everything including math and logic exist before God or because of Him? What we know from Scripture is that God has always been and I think actually to take that beyond that notion is to begin to be distracted. What I want to focus on is that Scripture paints God as eternal. Eternity past, eternity future and that is our focus. Everything that is in the mind of God is what was in the mind of God and He doesn't give us a lot of information beyond that. But what I want to know is that I serve a God who never had a beginning point and never will have an ending point.