True, there are natural events that cause suffering. However, those natural events also often have creating impacts as well.
If we look at Genesis, the first creation story, what is portrayed is that God created out of chaos. When God began creating the earth, there was chaos. In order for creation to be possible, God took chaos, and organized it. After each separation that God did (light from dark, water from ground, etc.), it is stated that it was good. Thus, the idea that is being portrayed is that chaos is also good. While the story may be mythical, it does present a "truth," that chaos does not have to be negative. It allows for the creative process.
When natural "evils" occur, often they build up. The beautiful geography we have in parts of the United States comes directly from these natural "evils." Not to mention the land that is formed from that natural evil that is volcanoes. Even something such as a forest fire, which can cause much suffering, has a positive effect in that it ends up allowing that forest to become healthier.
As for the possibility that one can be unable to suffer, yet humans still have a tendency to do evil, that really isn't logical. If humans were unable to suffer, and I happened to kill a person's entire family, is that actually evil? It did them no actual harm. There obviously wasn't any love there, or any real caring, as the family didn't suffer from the loss. An event like the Holocaust then wouldn't be a major problem, as no one suffered, no one really loss anything, and the logic behind it would have been for the betterment of all people. Evil really wouldn't be there.
Suffering does not have to be a bad thing. Victor Frankl, in his classic work, Man's Search for Meaning, details how suffering can be a positive act (not that we should search for it, but that it doesn't have to be all negative). Suffering can greatly help shape who a person is. It can cause them to become that much stronger. Frankl was no stranger to suffering either, as he had experienced life in a concentration camp.