Not vaccinated.. uh oh. I would be losing sleep over heartworm.. I have seen personally (through a friend who does rescuing) what heartworm does to a dog. I do think that it;s a great thing that your cousin is not giving you a hard time about keeping the dog; he does not deserve it, if it is not being cared for and protected against disease.
If it were me, the dog would go asap. I am very territorial about our place myself, and could not stand the thought of an untrained dog running amok.