What are the ruling(s) about organs separated from the body before death? For example amputated limbs. Can they be treated as rubbish? Or should they receive proper burying?
Please explain whether the ruling(s) cover clipped nails, hair, etc.
They should be buried in respect to the human being. But it is not from Shariah that people wash these legs, pray on them, or follow their funeral. These legs should also not be kept until the death of the person from whom they were taken and buried with him.
Same will apply on all body parts
Now on your second question, that if same is to be done with nails etc:
From Fatwa on Disposing of human body parts after some surgeries:
Pulled-out teeth and molars are not included among the human body parts that must be buried, because they fall under the ruling on parts separated from the body, and do not cause any harm if not buried.
Same will apply on hair nail etc.
You didn't demand any reference in your question, so i have not looked up any. I looked up the rulings and you can read through them for further reading.