As a transplant surgeon in training, it's fairly obvious where I stand on organ donation.
To address Daniel Ritchie's questions specifically:
1. This is a complicated question, because there are many variables. For people who consent to organ donation while they are still alive, it seems perfectly reasonable to follow through and take their organs when they die. But what about people who die without formally signing a consent when they were alive. Does the family (or close friends) have the right to consent to organ donation? If there is no family, can the organs be taken without consent at all? And the big question, exactly what constitutes death? It would be interesting to discuss these questions in detail, but these can be very complex issues that are still being debated in the transplant community.
2. On a moral level, it seems that there is nothing immoral about donating a single kidney, part of a liver, or bone marrow as a living donor. I can function just as well with a single kidney, so I would be perfectly willing to donate one to help someone else. Likewise for a partial liver donation. I see nothing in the Bible that would indicate this is wrong, and since it preserves life of another person I see it only as a positive, selfless act. Likewise, I see nothing wrong with consenting to organ donation after we die. Again, we no longer have need for a liver or kidney or heart when we're dead. Why not save someone else with our now useless organs? But the question becomes more complicated when you talk about paying people for organs, advertising for organs, etc. The reason is because it creates a fundamental inequity in the system: wealthy people, even if they aren't as sick, might get directed organ donations earlier because they can pay people for direct donation. Some people would argue that this is immoral, others would not. Also, is it wrong to donate organs if you're paid for them?
So, your questions can be very complex, but I would argue that as a basic level organ donation is a positive thing, and does not violate any Biblical concept. I know it's not the thread topic, but more complicated is the issue of who should
receive the organs, and on what basis.