Organs that exist twice (e.g. lungs, kidneys) can be donated.
All other organs cannot be donated because their removal leads to the death of the person.
It doesn't have to just be that they exist twice, but, rather, according to the second principle you stated, the organ removal cannot cause the death of a person. You can, for instance, remove part of a liver. You could, in theory remove both eyes, etc.
Apart from this moral principle, however, I find something
contra naturam to transfer organs from one person to another. These are more than just lumps of tissue, as I hold that they're permeated somehow by the animal soul and are tied to the individual who was born with them. I've heard stories from, say, eye transplants, where the organ recipient could see some things that the donor had seen in his life. With that said, if someone needed a kidney to live or at least to get off dialysis, I don't think it would be immoral to donate one.