What is the ruling on disposing of a duck of unknown ownership and its eggs, what must be done to make it permissible, and what should be done with the eggs it is currently sitting on?

It is not permissible for your brother's friend to dispose of the duck by giving it away if it is highly probable that it belongs to someone else, due to its presence in inhabited areas. It should be treated as a luqatah (found item). If it is highly probable that it does not belong to anyone, then it is permissible to dispose of it, as it is game.

If there is doubt about it, it should be treated as a luqatah. The basic rule for a duck found in inhabited areas is not to dispose of it before announcing it for a year. Its finder can donate it as sadaqah on behalf of its owner after announcing it for a year. If its owner comes, he is given the choice between approving the sadaqah or getting a similar item in return. Its eggs belong to its owner, and there is no harm in benefiting from them, but the finder is the guarantor. However, if it does not belong to anyone, there is no problem in the permissibility of disposing of it.