Does a daughter inherit her deceased father, knowing that she was born from an unregistered customary marriage, and is she entitled to bear only her father's name without inheriting?

If a common law marriage fulfills the conditions of a valid Islamic marriage, namely the presence of a guardian (wali), witnesses, offer (ijab), and acceptance (qabul), then it is considered valid Islamically, and the daughter resulting from it is attributed to her father and inherits from him. If the marriage is invalid due to a missing condition, paternity is still established, and inheritance likewise, if the husband believed in its validity. Paternity and inheritance are also established if a Sharia judge rules on its validity.