You can try looking on popular manga reading websites like MangaDex or MangaReader. Sometimes, libraries also have digital collections of manga you can access.
Yuri hurt fanfictions are great for character exploration. When Yuri is hurt, whether it's physically or emotionally, it's like a magnifying glass on his character. Physically, his pain tolerance and his love for skating are highlighted. He'll push through the pain to get back on the ice. Emotionally, we see how he processes things like rejection or disappointment. His character is complex, and these fanfictions peel back the layers. They show his growth from being a hurt individual to someone who learns and becomes stronger, and his relationships with Victor, Yurio, and others play a crucial role in this exploration as well.
No, it's not common in the context of the real story. Characters exist within their own world in Yuri on Ice and don't typically engage with fanfiction which is something created by fans in our world.
It could be quite an interesting experience. They might be surprised at the different interpretations of their relationships and stories. For example, Viktor might be amused by how some fanfictions exaggerate his charm. Yuri could be either flattered or a bit embarrassed by how his insecurities are explored in some fictions.
You can try Archive of Our Own (AO3). It has a large collection of fanfictions, and you should be able to find some that combine Harry Potter and Yuri on Ice.