The “problem” of evil
Atheists think that the “problem” of evil is proof that God does not exist. The argument is that if God is good there would not be evil in the world but there is evil in the world like tsunamis so God is not good but God is supposed to be all good so God does not exist.
But think about this? To say that there is a “problem” of evil is to say that there is evil. If there is evil then there is good. If there is good then there must be a standard of good. That standard of good cannot be human because humans are not good so the standard is God. So when you ask about the “problem” of evil you actually prove that God exists.
I don’t see anyway around that.
There are a number of points where I think you are incorrect:
1) The problem of evil is not proof that God doesn’t exist. Rather, it is, at best, proof that a particular characterization of God is logically self contradictory.
2) You do not draw a distinction between moral and natural evil. When someone does something bad, that’s a moral evil; a tsunami is a natural evil. There is no evil intent behind natural evils.
3) If not everyone agrees about what is good, then there is no overall standard of good.
4) You don’t have to be good to have a standard of good, just as you don’t have to be beautiful to have a standard of beauty.
For all these reasons, bringing up the problem of evil does not, in any way, validate the existence of God.