What does the Sharia require regarding the aforementioned inheritance issue, and does the questioner's father have a right to half the feddan his grandmother promised him?

If what is mentioned is taken as given, then the half feddan is an inheritance from the grandmother to be divided among all her legal heirs according to Islamic law. Your father is not exclusively entitled to it merely because of the grandmother's promise of a gift, nor is your mother entitled to it merely because it was written in her name, as long as the grandmother did not intend a gift by that writing, but rather a threat. It becomes your mother's if the grandmother gifted it to her as a gift that fulfills the legal conditions. Similarly, the check written to the paternal aunt is an inheritance to be divided among the heirs according to Islamic law, and the aunt is not entitled to anything from it except her legal share, because merely writing the check does not constitute a valid gift. If there is a dispute, the Sharia (definition) court is the reference.