shes come from being constantly yelled at and orderd around to a loving family & Im just scared of nagging at her,
Don't be afraid to set boundaries & rules though. You can still do this lovingly.

Kids / teenagers need some boundaries, it gives them a safe space to be in and also gives them something to push against - that might sound silly but I am a strong believer that it is teenage nature to test boundaries & see how far the rules will stretch, and it is better for everyone if those guidlines are there. If you draw a 1msquare, they will go to 1.5m. If you go to 2m, they will go to 3. But if there are no rules at all, there are no limits & they will feel lost. Does that make sense?

Rules need to be consistent though, and fair. And maybe inviting her input will help. See what she thinks is fair & reasonable. I read something once, by a parent who allowed his teen to choose his own punishments. They were usually more severe than what the parent themselves would have chosen. Likewise you might find that if she has input into her chores she's more inclined to do them.
No, she is absolutely not too young for a chore chart.
Also it's ok to cater for likes & dislikes IMO. My DD1 hates the dishes, DS1 hates hanging washing. Chores are often alloted with this in mind.

As for the hair cut, that's betwen you & your mum. Your mother might feel that she is still the mother & should have a say in these kind of things?
My personal opinion is that by 15 (in fact for us, it was 13) they are more or less able to make their own choices about hair. It is such a big form of self-expression, and much less permanent than piercings or tattoos. We let them go for it in that area. But like I said, that one is really between you & you rmum - you might need to talk with her too I think & establish exactly what you are responsible for if your sister is living with you.