Short answers, pulled from the story.
Batomorphi is a division of cartilaginous fishes commonly known as rays. The 5th edition of Fishes of the World classifies rays under this division name rather than the alternative designation Batoidea. Along with the shark division Selachii, Batomorphi composes the subclass Elasmobranchii.
Batomorphi includes well over 600 species spread across 26 families. This makes rays the most species-rich group among all cartilaginous fishes.
The oldest confirmed ray is Antiquaobatis, found in deposits from the Pliensbachian stage in Germany. The Pliensbachian falls within the Early Jurassic period.
Bottom-dwelling batomorphs take water in through spiracles on top of the head and pass it outward through gill slits on the underside of the body. Most fish breathe by drawing water in through the mouth, and sharks have gill slits on the sides of the head.
Mermaid's purses are the leathery egg cases produced by oviparous skates and some rays. They are commonly found washed up empty on beaches in areas where skates are abundant.
A 2021 study published in Nature found that oceanic sharks and rays declined globally by 71 percent over the preceding 50 years. Overfishing has pushed three-quarters of these species to the point of threatened extinction.