If it were legal to buy and sell organs, the organ shortage would go away overnight. People would have more of an incentive to donate their organs when they die to help their family financially.
All of these, however, are much easier questions to answer than setting up a government bureaucracy to decided who deserves the organs or who deserves to donate.
You could start it out very simply-- if you are a donor, your estate will get x$ toward your funeral or burial expenses if something is harvested. Even a capital criminal who cares about nobody might be persuaded to donate with the promise of a nice tombstone versus a brass marker.
That's a great point. It could still be something that is a nominal level of cost to a recipient and their insurance to the extent of covering all funeral costs plus a nugget (say +/- $10,000 for an organ).
I figure you don't want to make it too valuable otherwise there will be people/relatives "speeding up" the death schedule.