About 1) the situation is actually comparable with the way inheritance is handled.
Most people specify what to be done by issuing a will, but there is not unlimited freedom there. For example, in some countries it is difficult to completely disinherit someone. If there is no will, your property is split in some way among your relatives. The default option, if there are no heirs, is that the inheritance falls to the state.
After death your organs likely won't be able to benefit your relatives, unless they are currently in a life-threatening condition themselves. Dunno how long organs can be kept around, but I doubt that there is no expiry date after which they have to be tossed out. In that case, the organ would have gone to waste by sitting in a freezer.
Therefore, treating your dead body's organs similar to inheritance would make it moral to pursue the default option in most cases, which is donation to the health system. Still, there is the possibility to opt out, which is an extra effort similar to (but way less involved) than issuing a will.