How would the government know whether the liver A for patient B came from a legitimate donor, some unfortunate homeless person within our country who was robbed of his liver against his will, or from some foreign national who lives in one of those totalitarian and/or lawless countries where human life isn't worth much ?

If it didn't come from a legitimate donor, either the doctor has to explain why he didn't say who the donor was, why the donor doesn't know about it, or why the "donor" still has a whole liver.

As for the foreign country, just don't accept organs shipped across national lines if the other nation doesn't check the stuff. I suspect it's a bad idea to ship organs across national lines anyway, as opposed to just shipping the donor, but I don't really know all that much about how organ donation works.

Would it be worthwhile to shorten someone's life by, say, 10 years (due to their loss of a kidney), in order to grant someone else 5 extra years of life (by using that kidney) ?

No, but I'm willing to bet that's not what the ratio will be. If losing an organ hurt you more than it helped the person getting it, nobody would ever consider donating them.

Edit:

Furthermore, let's say that people could sell their organs after they die -- by proxy, presumably -- as per your scenario above. Doesn't this create a powerful incentive for unscrupulous agents to speed their demise ?

You mean their next of kin? I don't see how it's much more of an incentive than it would be to kill them just for the inheritance. Also, I suspect that the health insurance company would normally take the money, since most people aren't going to be in a position to donate their organs and people tend to be loss-averse.