Short answers, pulled from the story.
The name Skaði appears alongside words like skadus and scato that mean shadow. Scholar John McKinnell suggests this points to a personification of Scandinavia or an underworld figure. The word skaði also exists as a noun meaning harm or damage in Icelandic and Faroese languages.
Skaði appears in poems compiled during the 13th century from older traditional sources. She features in Grímnismál where Odin describes her living in Þrymheimr after her father's death. Her presence is confirmed in Skírnismál, Lokasenna, and Hyndluljóð which establish her identity as a jötunn and goddess associated with winter hunting and mountains.
Their marriage failed because she wanted to live in mountain home Þrymheimr while he preferred sea-side Nóatún. Njörðr complained that wolf howls sounded ugly compared to swan songs during nine nights in the mountains. Skaði replied that gull screeches kept her awake on the sea beds before they eventually split up.
Skaði places a venomous snake above Loki's face so its poison drips directly onto him. His wife Sigyn sits beside him holding a bowl to catch the dripping venom until it fills. When the basin fills with poison Sigyn must leave to empty it allowing the acid to burn Loki's face again causing earthquakes.
Chapter 8 presents Skaði as a historical queen who first married Njörðr but refused to have sex with him before marrying Odin instead. The text states that Skaði and Odin had many sons including one named Sæmingr who became king of Norway. This euhemerized version transforms the goddess into an ancestor of Norwegian kings within Snorri Sturluson's history.