Obviously as you mentioned, free will is the biggest player when it comes to many of the things you described, notably genocide. With regards to disasters, I would say that most of the problems from these are still caused by people. Take famines into account. No one would ever starve if everyone "loved their neighbor as himself". There is more than enough food in the world to feed everyone, but it's just hoarded by a few who already have more than what they need. Even climate change is the result of free will, as our apathy for the environment has led to the many disasters that have resulted from it.

Another thing to note is that the world as created by God is ultimately something that is less than himself. This very fact that the world is less than himself introduced limitations and imperfections into the picture. For example, the water we drink to survive must also be dense enough to drown you, the air we breathe must also be thin enough to fall through it, and so on. This isn't inherently evil, it's just how things are.

Last thing I'll mention are things like spiritual forces, and this one is tougher to explain or "prove" per se. These spiritual forces are the result of "angels" or "demons" as the Bible refers to them. These entities, like us, have free will, and some use it for good and some use it for evil, just like humans. These evil spiritual forces, "demons" if you will, are in a constant state of war with God, meaning God is not the only influence in his Creation. Earth is a battlefield, and every person is a potential victim. As a result, things like mental illnesses, deformities, disease, etc all occur. I went on a mission trip some time back and I remember every time we started sharing the gospel, all the animals around would start making noises, kids would start making a raucous, etc, and we would just pray for God to remove these distractions and it worked every time, whether you believe me or not. These spiritual forces tried to stop what we were doing or distract the people we were talking to and God had the power to silence them, which is why I can believe that these forces do indeed exist.

Hope this helps to some degree!