Why does God sometimes stop helping sinners and even give them over to their sin?

You did a pretty good job yourself, so there is not much to add. It started in the Old Testament: right before the flood God said he would give them 120 more years, and during that time Noah built the ark, and when the 120 years came they all died in the flood. This is the thing about dealing with God: it is all or nothing, there is no in-between, no little corner to hide in and be left alone. You have to deal with God or the devil, and when you deal with God he says, you either take all of me or you can have nothing of me; if you are lukewarm I will spit you out of my mouth. He is serious; he is not some milquetoast granddaddy in the sky, which is why he punishes so readily throughout the Bible, to give us the message that he is not fooling around. So if it is all or nothing and you deliberately break his laws and stick it in his face, then, as you listed, here is what he is going to do: he gives you over to your sin. But we also have a very merciful God: if you sin against him and later repent and say sincerely, I do not know what came over me, I am sorry, he promises to accept that, because what he is looking for is someone sincere who wants to be rid of their sins and to be like God, who is holy, holy, holy and righteous, righteous, righteous.