I think this girl is in pain, and that's why she's being a pain.
Her mother abandoned her. She has a new guy, and a new baby, so she dumped her daughter and bolted. I can't imagine what it must feel like to think that your own mother doesn't give a hoot about you, especially when she obviously cares enough about your sibling to take them along. So this girl is very hurt and angry, and she's taking it out on any adult within reach. Since the OP is the mother's sister, she makes a particularly good stand-in for the mother that this child is really upset with. I'm wondering if the reason the niece blew up about the stuff for the Salvation Army, is because she somehow identified with the things that were being given away, sort of like the way she feels that she herself has been discarded?
I suspect the grandparents aren't thrilled to be raising their grandchild, either. Older folks simply haven't the energy that dealing with children requires; I'm sure they were thinking that their child-rearing days were over. (I suspect they also have some issues with the mother who would abandon her child like that). It seems to me that sometimes, the G-parents are treating the OP like a child ("you need to play nice with your little sister") and at others, are expecting the OP to parent the niece for them.
There definitely needs to be some ground rules set here. A lot of it has to do with treating each other with the respect they deserve. Absolutely, the OP has the right to set boundaries, with the child and with the grandparents. "I do not deserve to be spoken to like that, and I will not be treated that way" are things that she is well within her rights to say. I don't agree with "punishing" anybody in this situation, but there should be a clear understanding that actions have consequences, and violations of the very sensible rules will have very well understood consequences.
I'm curious, where is the OP's husband in this? Maybe, being a "disinterested party", he can help to spell out the rules in a way that will seem a bit more fair to all parties. (though I can't blame him if he just wants to stay out of it, either!)