>No, we are judging what is claimed about this God.
Well, part of what is claimed about God is His Infinitude. He is infinite and all mighty. An infinite and all mighty God can do what He wants. He answers to no one. That is the ultimate answer to your questions. Furthermore, His character does not provide a reason to believe in Him or not. He could be the heartless, vile god that people try to strawman Him to be, but that would not be a reason to not believe in Him and give Him worship if the evidence was there that He exists.
God's existence cannot be proven outside of Scripture. I believe that there is sufficient historical evidence behind the claims of Scripture - chiefly those around the life, death, and resurrection of Jesus of Nazareth - to provide a reasonable foundation for the Christian faith. I would recommend reading this book. It contains the transcript of a debate between Antony Flew, an at the time atheist philosopher, and Gary Habermas, a Christian apologist, and a few other essays. In it, Flew admits that the only reason he doesn't believe the evidence for the resurrection is because he a priori decided the resurrection cannot happen.
>No, I don't want to say it is irrational for other people to believe in this. It seems to me it would be perfectly rational for them to believe in this, but I can't cope with this idea at all. It seems to me so unlike anything else that happens in the universe.
Editor's emphasis. If you are going to be an honest skeptic, you have to address the actual historical evidence as historical evidence, as Dr. Flew did. He dismissed it based on an a priori assumption, but he does address the evidence.
Admitted Christian here. It always feels like the elephant in the room is that arguments like this are a veiled critique of Christianity, so that's how I am approaching this. We actually just discussed this in my philosophy class, and as a Christian, I've looked into it theologically as well.
One of the biggest problems with the problem of evil, at least as it is presented here, is that there is one glaring assumption that doesn't get discussed: that I am not evil. I being whoever is making the argument. From a Christian perspective, everyone is evil. It's also pretty obvious that no one is good. If you don't want to go so far as to say you are evil, okay, but you aren't good. So if a good god must destroy evil, then he must destroy everyone. But he hasn't destroyed me (at least, not yet). Why? I'll get to that in a moment.
It also assumes that God is not involved in either a) suffering or b) a response to the evil in the world. Assume, for the sake of argument, that the Genesis account of creation is true. Genesis 1 and 2 describe the creation of a good world and a very good man and woman. I think we should understand that to mean perfect. In Genesis 3, we see the Fall. The serpent (nearly universally understood to be Satan in some way, it is also possible to understand serpent as dragon) tempts the woman to eat of the tree of knowledge of good and evil, and she convinces the man to eat of it as well. This was the one thing they were commanded not to do on pain of death.
But then God comes into the garden, gives them a chance to repent and ask for forgiveness, but when they don't, He curses the serpent, woman, and man. The serpent's curse is that the woman's offspring will crush his head even though he will bruise his heel (3:15). The woman's curse is pain in childbearing and ill-content with her womanliness (3:16). The man's curse is that the work he was given to do is now cursed; thorns and thistles will come forth from the ground, and by the sweat of his face will he eat his bread. And finally, death will come to us all. (3:17-19)
So we are all cursed by God for our original sin, and our own actual sins that we commit cause further suffering. Is God doing anything to respond to this suffering? That's what the serpent's curse is all about. The Seed born of woman will crush the serpent's head. This is the promise that God gave Adam and Eve in the Garden to restore faith to them. That a singular man will one day defeat Satan and all that he has brought about. When the time had fully come, God sent his Son, born of a woman, born under the law, to redeem those who are under the law.
God dealt with evil in the body of Jesus of Nazareth hanging dead on the cross and then resurrected three days later. That is how the Christian would respond theologically to the problem of evil, but then there's a whole host of questions about that central truth claim: Did the resurrection happen? I certainly think there is plenty of evidence to suggest that it isn't irrational to believe that it did.
Dr. Antony Flew would disagree. Check out Did the Resurrection Happen?.