"Evil" takes at least of a couple of meanings. One is just 'hard things', like suffering or catastrophe. But "evil" is also a reference to sin, or sinfulness, or even the horror behind sin. We are talking here about this second meaning: Sinfulness. I'm not myself entirely satisfied with the easy out, though I believe it —that sin does not "exist" as such. The idea is that it is not of itself a "thing", but is only, as theology carefully puts it, "the privation of Good." I believe that much is true, but I don't think that is the whole explanation.
You speak of God wanting. If God exists —that is, if omnipotent first cause exists— pretty much everything we can say about him is by way of anthropomorphism. God, according to excellent theological philosophy, in the attribute called Divine Simplicity, is not made of parts. He doesn't have passions and desires the way we do. That is not to say he doesn't have passions or desires at all, but that they are not parts of him. He is not now ruled by anger and later not. His anger is not without his mercy. And so on. We just don't know how to speak of him without dealing with his various attributes separately. And even his passions and desires are not at all like ours. We just don't have adequate words to describe them.
I don't know if you have read much of philosophy of language and related subjects dealing with how we think. One phenomenon that is apparently intrinsic to our thinking, is that we are not just reasoning, but that we are reasonable —in other words, that our words make perfect sense, and are capable of well-describing what we mean. We attribute substance to our words, when in fact our words do a poor job of describing our concepts, and our concepts are necessarily short of knowledge, understanding and meaning.
So, no. God did not make the world with evil included. He made it so that it would produce evil, specifically, made it so that his angel, Lucifer, would rebel. This was not a mistake, but was intended. Yet God himself did not create that evil. Lucifer began it. Satan is the original author of evil. God caused it, by means of Satan causing it. God "wanted" it? I wouldn't say so, but that he intended and in fact caused it to be so. We have some passages of Scripture that come right out and say so. But also, common sense demands it. As you say, he knew it would happen, yet created precisely would result in it happening.
And so, yes, God deliberately chose a system where he would have to punish Adam and Eve, and destroy the world in a flood, and send himself to be sacrificed to himself in the form of Jesus. God is magnificent, but he is not like us. We can't understand him by thinking of him how we think of ourselves. If he did not know it was going to happen when he created, then he is not after all, God, Omnipotent First Cause, but just a super-human, a god.
