Your questions presume a certain preconception of God, as something that ought to intervene to mediate suffering or discomfort of humans, but that no such God intervenes neither proves nor disproves existence of God. It just means that if there is God, it is not one that necessarily intervenes; or if there is sometimes intervention, then it is inscrutable.
If the ecosphere develops so as to provide potential for something, then there is a likelihood that it may exist. Mosquitoes, as well as ticks and vampire bats, exist because there are warm-blooded mammals that can provide energy by taking some of that blood. The diseases we get from them come to us because these bloodsuckers provided useful hosts. We call life's creating niches. Leukemia exists because errors occur in DNA replication at a regular rate, but then also so do all other forms of life exist for that reason as well. The mutations provide possibilities, but the possibilities provided are not necessarily those preferred by the particular organism in a particular time and place (for example, sickle cell disease allows a population to flourish, but with the sacrifice of a particular percentage of its members for whom its consequences are tragic. Can say the same thing for war, which we often into enter into with patriotic alacrity, even glorifying the tragedy that occurs when it serves our purposes.)
Worship of God does seem to indicate what is called God exists as a force of some sort, but not necessarily in the terms by which people worship it. Belief in God provides solace in the face of pain and suffering, even if there is no God that actually intervenes. Many believers try to explain your answer by saying that God gives us suffering so we can learn to transcend it, but such transcendence can come without a belief in God too -- for example, a mountain climber overcoming adversity encountered in climbing a mountain, while the risk of adversity a person allows him/herself gives the rest of us sense of the expansiveness of the human spirit and the powers of the universe when a particular climber gets killed while trying to overcome such adversity.