I am fuzzy on the details but I know in the book a sorcerer has a white haired girl with a black cat that is his slave. There is also a girl with long red hair that climbs a sacred Mountain, and her father is forced to cut off her hair and dye it to make people think she is a boy, because climbing that mountain is punishable by death for females. The red-haired girl is also badly burned, especially on her hand.
This is Sorcery Rising by Jude Fisher.
I don't have a good summary to provide you, but it starts with an apprentice stealing the last bit of magic from their master, who plans to eliminate magic to save the world. There is a catlike familiar, who is actually a sort of a demon. The action then switches to a knife-maker's daughter who likes to climb mountains and indeed had her hair cut to hide her from the religious authorities.