It sounds to me like the problem it with your own self hatred and insecurities which you are projecting onto her.
I think you need to concentrate on why you feel like you do, what problems you have which are causing it rather than on anything she does.
At the end of the day, you can't change her, you can only change you, you feeling, reactions etc...
It might not be a particularly pleasant thing to do, no-one likes realising they have an issue, but look to the possible long term situation, you could be a happier you and you could potentially have a happy relationship with your sister so could be win win.
No pain, no gain as they say.
It seems like a lot of your communication is online where things can be grossly misinterpreted. Why don't you try and spend more time together in person?
Even happy people have their problems, they just maybe have different ways of dealing with things but it's easier to look at things with a grass is always greener view.
If you've caused her problems in the past, could it be she's trying to keep an eye on you?