Why do we suffer because of Adam's sin?
We suffer for Adam’s sin because we inherited this depraved state from him. Adam and Eve had free will and God gave them an opportunity to exercise that free will with one rule about not eating fruit form a single tree in the garden. They choose to exercise that free will to disobey that only command. They entered into what we call the fallen state where they took on a rebellious sin nature. In that state they conceived the rest of humanity and in a sense, we are Adam. Animals and human reproduce of their own kind. When they reproduced the rest of us it’s as if we are saying that we have that same nature and that the entire human race sinned along with them. Romans 5:12 says all men have sinned. Notice it is past-tense. It’s as if we all sinned at once. Notice, also, that Ephesians 2:1-2 calls us the sons of disobedience.
Biologically, we are a family and, in a way, we are one. We all have Adam and Eve’s DNA floating around in us. In a very real way, I was there with Adam when he sinned. I had just not spawned yet. Millard Erickson puts it this way,
“The entirety of our human nature, physical and spiritual, material and immaterial, has been received from our parents and more distant ancestors by way of descent from the first pair of humans.” Augustine called this our seminal nature. John Calvin said, “By the corruption into which he [Adam] himself fell, he infected his whole seed… Our nature is not only utterly devoid of goodness, but so prolific in all kinds of evil, that it can never be idle.” 
I think we struggle with this because we look at this from an individualistic point of view. We need to look at it as if we were a tribe or, in more modern terms, we need to look at our corporate choice and condition. A number of biblical passages give us a look at the concept of a corporate identity.  We may object to this corporate view of sin and guilt because we feel we never had the chance to not sin. However, that does not mean that we should not be guilty because, in a sense, we ratify our guilt by our own sin. It is true that, by one man, we have all sinned but the good news is that, by one man, Jesus Christ, all men can be saved. “For as through the one man's disobedience the many were made sinners, even so through the obedience of the One the many will be made righteous.” 
In a way, this corporate view cuts two ways. In one way, we are guilty even before we are born. The upside to this is we could never be good enough on our own and by the life, death and resurrection of the second Adam (Jesus Christ) our debt is paid for. Yes it works both ways but the benefit is all ours.
What about pain and suffering caused by natural disasters?
Suffering from natural disasters is a result of Adams sin. God cursed the ground for Adam and Eve’s sin.  Suffering caused by all natural disasters can be accounted for as a result of Adam’s sin.
Hugh Ross also makes the argument that natural disasters are needed in order to make our planet livable. Without plate tectonics and volcanic activity, our planet would not be able to sustain life. We cannot live on this planet without natural disasters. In a very real way, the catastrophe is a small price to pay for a greater good.