In such a case, is it permissible to marry a girl who has been sexually assaulted by her father, especially with the absence of a guardian and the possibility of Islamic centers undertaking the marriage contract, and do weak hadiths such as "the green of dung" and "lineage is an infiltrator" apply to her, despite her religious and moral commitment?

The fundamental principle is that it is not permissible for anyone other than the closest guardian (wali) to solemnize a marriage. However, if it is impossible to reach the girl's father, and she has no other guardian, then it is permissible for the ruler or the Islamic center to solemnize her marriage, especially if there is no Sharia court in the country. It is recommended to marry a righteous girl, and the poor character of her father should not be a deterrent, as Allah Almighty says: "And no bearer of burdens will bear the burden of another."