Short answers, pulled from the story.
A Baugrygr, also called Ringkvinna, was an unmarried woman who inherited the position of head of the family in Scandinavia during the Viking Age and into the Middle Ages, usually from her father or brother. She held all the legal rights and duties associated with that role, including the right to demand compensation for the killing of a family member.
Under the Icelandic Grágás and Norwegian Frostating and Gulating laws, an unmarried woman reached legal majority at the age of twenty. She could choose her own place of residence and stand independently before the law in all legal matters. The one exception was the choice of a marriage partner, which remained a family decision.
Only the daughter or sister of a deceased man could inherit the position of family head and become a Ringkvinna. A broader category of women, including a paternal aunt, paternal niece, and paternal granddaughter, could inherit property and were called odalkvinna, but the headship of the family itself was restricted to daughter or sister.
When a Ringkvinna married, all the legal rights she held as head of the family were transferred to her husband. The sources that describe her authority make this condition explicit, and it applied to her full suite of powers, including the right to receive compensation fines on behalf of the family.
The Ringkvinna’s rights gradually disappeared after Christianization and are no longer mentioned in any law texts after the late thirteenth century.
The Ringkvinna is primarily mentioned in the Icelandic Grágás and the Norwegian Frostating laws and Gulating laws. All three bodies of law phrase the relevant provisions in much the same way.