With regard to the other evil, natural evil, all the arguments ultimately hinge on one premise:
There is no reason that justifies God’s permitting evil.
The problem with this premise is that it not only concludes that we know as much as God, it also concludes that we can or should be able to know as much as God.
If there is a God, I would expect him to know more than I do in every respect. A God who knows no more or little more than I do is not a being worthy of being called “God” (or even “god”).
This means that God, if he exists, must know more than I do or he is not God. If he knows more than I do, then there may very well be a reason why he would permit natural evil (perhaps natural evil even brings about a greater good), even though I may not understand how.
Just as moral evil cannot be used against God because he gave us what we really desire: free will (God cannot be held accountable for the use of our freewill which means we cannot use human moral evil to deny his existence), so natural evil, while repugnant to us, may very well have a reason, perhaps linked to a greater good, that we are unable to fathom because we do not understand things on God’s level.
Since we would expect a being with the attributes God possesses to understand things on a higher level than we do, and since we cannot expect a being such as he is to submit himself to inferior measurement (just as we would not submit ourselves to such), we cannot use natural evil to deny God’s existence.
I know I’m leaving quite a few steps and details out but this is a summary, not an exhaustive essay or paper.
This does not prove God’s existence (nor is it intended to). This also is not a theodicy (nor is it intended to be).
You must find other grounds to deny God’s existence.