Why are we punished for the sins of others, e.g. Adam and Eve and children for their parents' wrongdoing?

We are told that no one will be punished for another person's sins. However when Adam (as) and Hawa (Eve) went against Allah (swt) instructions and ate from the forbidden tree, they were sent on Earth as a punishment and test. Now, the rest of mankind did not eat from the forbidden tree, then why are we on Earth and being tested if this was a punishment for their wrong doing? Also if a person does something evil, we are told we will always pay for our wrong doing (unless we repent). We might pay for our wrong doing in this dunya or in the hereafter. There are many ways we can be punished in this dunya, e.g. through our wealth, health or even children. Now, why are the children being punished e.g. by being afflicted by a disability because of their parent's wrong doing?