Thank you to @TheLethalCarrot, without whom I would not have found this book.
The book in question was Half-Human (2004), which did indeed include part of Dusssie by Nancy Springer.
A girl who hears the sea amongst the busy streets of Manhattan. A princess born with the strength to clutch a sword hilt. A winged boy who cannot fly. In these stories, the kind where trees can become men and girls wake up with snakes for hair, people are not always what they seem. But they still must discover who they are.
Compiled and edited by Bruce Coville, with contributions by Tamora Pierce, Gregory Maguire, Jane Yolen, Nancy Springer, D.J Malcolm, Janni Lee Simner, Jude Mandell, Tim Waggoner, and Lawrence Schimel.
The three bolded parts of the first paragraph are the three stories I described in my original post. The first is called Water’s Edge (by Janni Lee Simner), and is about a young girl who finds out she is a Selkie. The second, about a young man with wings who can’t fly, is called Soaring (by Tim Waggoner). And the third, which became the book Dusssie, is called Becoming (by Nancy Springer).
This book was published by Scholastic in 2004, which also explains why it was in my fifth grade classroom.