While the sending of Adam and Hawa to Earth from their initial place of residence may be a punishment for the two, the birth of the rest of the humankind in this world is not a punishment for them.
The problems, suffering, pain we experience due to others' actions or choices is not a punishment for us, it is a natural consequence. E.g. you drive in the red light and hit someone and kill him. It's not a punishment for the person you kill. The punishment for everyone's acts will be decided and distributed in the judgment day without any injustice (Quran verses about everyone being solely responsible for their actions 2:134, 4:123, 6:164, 17:15, 29:12-13, 35:18, 39:7, 53:38-39).
A note about Verses 29:12-13: The people who lead others to sin, will have the burden of their own sins + the burden of the sin of "leading others to sin", multiplied by the number of people they led to sin. They won't have the burden of others' own sins. Others' will have the burden of their own sins (7:38-39, 37:30).
In your example, it is not a punishment for the child, it is a test, also a natural consequence of the parents' actions which they comitted even though they knew the possibilty of the negative outcomes (e.g. marrying to close relatives, drinking or smoking, etc.). It is a test for the parents to have a child with disability/disease (one may also consider it as a part of their punishments for their own sins that caused the child to be born like that). But it is also a test for the child, a test to endure and show patience about the disability/disase in his/her life. In the judgment day (with the light of the verses I mentioned above), the parents will be asked of their actions that led the child to be born with disability/disases and, in my understanding, the child will also ask his/her parents why they caused that and everyone will receive their just judgment.
May Allah give us an easy judgment.