As an adult with AS, I can lay out the way my brain works for you. This may or may not apply to your daughter, but their are some similarities I would suspect.
First of all, sibling rivalry exists between the most normal, average children in the world, and while it is something that needs to be dealt with, it's also something every family struggles with. My sister (two years younger, in my case) and I sometimes fought wars and were sometimes great friends. We both deliberately pushed each others buttons as well. If she's only acting out in one particular way toward her sister, I'd suspect that before any other motivation.
As far as how I think, for me and many people with AS/HFA, our world is extremely logical with little regard to what others think of the decisions we make. After all, if my decision is made in sound logic, everyone else is obviously missing something something and their "wrong" thoughts and impressions are irrelevant and inconsequential. So during my school years for instance, if I could find a different, more efficient way to solve a math problem, I would disregard my teacher's instructions to follow certain steps and continue to solve it that way regardless of if I was continually marked down for it. My a means b and b leads to c, so there's no need for d, e or f no matter how many times I was told to do it that way.
Even now, someone else's emotional response is fairly irrelevant to me. It's not that I can't tell they're upset that I'm picking apart their argument or making a point their are incredibly opposed to - it's that I think it's more important make my own point to them so that either a.) I can learn something more about their point of view by making them explain it in more detail or b.) I can correct whatever fallacy they have so that they don't continue to repeat it. This drives my mother absolutely nuts, but it's a part of me I can't really turn off. I just hide it better than I used to.
So when you're correcting your daughter, I would suggest you explain your own logic in the decision, no matter how absurd it seems. If you lay down a rule, explain why it's important. And be prepared for her to challenge that logic if it doesn't mesh with hers ("Because I say so!" is perhaps the worst response if you want a kid like I was to change her behavior), and answer whatever problems with it she might have. "I can't get your DVD because I am cleaning up right now. When I am done, I can get it. If you want it right now, you can get it." Whenever possible, lay out the underlying rationale.
As for the apology thing, there's a very good chance she might not be sorry if she doesn't understand what she should be sorry for. If I hurt someone's feelings by telling them their play needs more rehearsal, I think I've done them a favor by making sure they get better for the future. Many things come out "mean" that are just really, really overly blunt. And for me, at least, it makes life easier - I hate when people talk around a problem or hint at something or disguise a critique in a compliment.
Following this, she is probably "manipulative," though likely not consciously. If she's like me, she learns what behaviors from her get the reactions from you and forms essentially an "if XXXX, then XXXX" reaction in her head. The fact that her tantrums stress you out is irrelevant, because she gets what she was looking for. In many cases, the best response to a tantrum is to a.) make sure she can't do anything ridiculously destructive to hurt herself and then b.) walk away. Or if she's got a room or something, then immediately put her in her room and leave her there until she's calmed down. Either go negative reinforcement or (if possible and IMO better) no reinforcement. She'll learn pretty quickly she has to find new ways of getting what she wants.
So to that - explain to her what you want out of her being "sorry" for something. It's obviously not just the words, and you can't force her to regret a behavior she thinks is completely appropriate. What you want to start to get her to do is to try and project herself into the other person's shoes (which is both really hard and really easy for me - I can "logic" out what a person ought to be thinking and feeling and am therefore pretty good at arguing both sides of a point, but I have a hard time understanding their emotional connection to an incident). Turn the tables on her for some of her behavior - have someone lay across her and pin her down and ask her if she likes that. The more she is able to understand how certain behavior makes her feel, the more she'll be able to "logic" out how it affects others.
A checklist/schedule is a great idea, though be aware they can also raise the stress level immensely on mornings where it has to be different (doctor's appointment or plumber coming over or whatever). I love my schedules, but I can become fixated on them too much and freak out when something unexpected happens.
