Is this woman permissible for him again after she has been divorced multiple times due to personality issues, or has the relationship legitimately ended, and she must marry another man?

Your relationship with this woman is not permissible. Since she has been divorced three times, it is not lawful for you to take her back unless she marries another man and he consummates the marriage with her, according to the Almighty's saying: "And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband." As for the daughter, if you believe that your relationship with the woman was lawful, then the daughter is attributed to you, and you must fear Allah regarding her. The care you provide for her is better for her than her grandmother's custody. It is advised to refer to the Sharia court.