Is it permissible for children to enter their maternal aunt's house if their father swore an oath of divorce that they would not enter it? And is that considered severing ties of kinship?

The father does not have the right to prevent his daughter from visiting her aunt unless there is a legitimate Sharia reason, because that may lead to cutting off ties of kinship. Obedience is only obligatory in what is rightful, and there is no obedience to a creature in disobedience to the Creator. As for the oath, if the girl enters her aunt's house, a divorce occurs according to the majority of scholars, whether the father intended to effect it or intended to threaten and prevent. Some scholars are of the opinion that if he intended prevention, the divorce does not occur, but rather he is obliged to offer an expiation for an oath. As for the second question, obesity does not prevent marriage legally or customarily, and a suitor should not be rejected merely because of his obesity.