What is the ruling on amending official documents for a family name, and what should a girl do if her biological father does not agree to give her his name?

Firstly, adoption in a way that attributes a person to other than their father and grants them the rulings of a biological child is impermissible. This is explicitly forbidden in the Quran, and this situation must be rectified.

Secondly, if your biological father married your mother, it is obligatory for him to attribute you to himself. Lineage is a right of the child, and it is not permissible to deny it except through li'an (oath of condemnation). If he denies your lineage, you have the right to take the matter to court. A severe warning has been mentioned regarding those who deny their child. If he did not marry your mother, the majority of scholars hold that you are not attributed to him, but rather to a general name such as "Abdullah ibn Muhammad." Some scholars have gone to the extent of permitting the attribution of the child's lineage to the fornicator, and this is the view of Shaykh al-Islam Ibn Taymiyyah.

Thirdly, when marrying, a wali (guardian) is necessary for the validity of the marriage contract. The guardian of a woman is her legitimate father, then the grandfather, then the brother, then the uncle. If there is no legitimate father, then the judge assumes the role of guardian for the marriage. We advise you to be dutiful to your mother and treat her kindly.