This is a good start. You portray Hermione's character well, especially when she doesn't want to use the time turner for anything but schoolwork. However, I'm not sure she would just burst into tears like she did with Hagrid. She seems more like the sort of person to hide her feelings.
I'm very interested to see how Harry and Ron treat Hermione later on in this fic.
Was that a Sherlock Holmes reference at the end? That's awesome :) I really liked this story! I love reading Hermione's inner dialogue, it was really interesting!