If a Jew has a non-Jewish father, and sells him his chametz, but then the father (unfortunately) dies during pesach, does the Jew then have ownership of the chametz? What can he do? Would he have broken a biblical commandment simply by the act of his father dying and he inheriting?

Shalom, A nice question, but a very clear answer. The famed Nodah B’Yhuda (ch. 18), writes that even if a Jew didn’t sell his chametz and dies on Pesach, his inheritors do not acquire the chametz automatically until after Pesach, and even then, the rabbis did not “punish” the inheritor to prohibit it as if it were “chametz she’avar alav haPesach”. How much more so, in this case that it’s allowed and there’s no prohibition involved. In addition, if you are asking practically, we don’t worry that the father may die during Pesach, so he can sell his chametz to his father. Chag kasher v’Same’ach Rav Ari Shvat